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Scientific notation

Scientific notation is a form of presenting very large numbers or very small numbers in a simpler form. As we know, the whole numbers can be extended till infinity, but we cannot write such huge numbers on a piece of paper. Also, the numbers which are present at the millions place after the decimal needed to be represented in a simpler form. Thus, it is difficult to represent a few numbers in their expanded form. Hence, we use scientific notations. Also learn, Numbers In General Form .

For example, 100000000 can be written as 10 8 , which is the scientific notation. Here the exponent is positive. Similarly, 0.0000001 is a very small number which can be represented as 10 -8 , where the exponent is negative.

Scientific Notation Definition

As discussed in the introduction, the scientific notation helps us to represent the numbers which are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent. The exponent is positive if the number is very large and it is negative if the number is very small. Learn power and exponents for better understanding.

The general representation of scientific notation is:

Also, read:

• Scientific notation formula calculator
• Scientific Notation Calculator

Scientific Notation Rules

To determine the power or exponent of 10,  we must follow the rule listed below:

• The base should be always 10
• The exponent must be a non-zero integer, that means it can be either positive or negative
• The absolute value of the coefficient is greater than or equal to 1 but it should be less than 10
•  Coefficients can be positive or negative numbers including whole and decimal numbers
• The mantissa carries the rest of the significant digits of the number

Let us understand how many places we need to move the decimal point after the single-digit number with the help of the below representation.

• If the given number is multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive. Example: 6000 = 6 × 10 3 is in scientific notation.
• If the given number is smaller than 1, then the decimal point has to move to the right, so the power of 10 will be negative. Example: 0.006 = 6 × 0.001 = 6 × 10 -3  is in scientific notation.

Scientific Notation Examples

The examples of scientific notation are: 490000000 = 4.9×10 8 1230000000 = 1.23×10 9 50500000 = 5.05 x 10 7 0.000000097 = 9.7 x 10 -8 0.0000212 = 2.12 x 10 -5

Positive and Negative Exponent

When the scientific notation of any large numbers is expressed, then we use positive exponents for base 10. For example: 20000 = 2 x 10 4 , where 4 is the positive exponent.

When the scientific notation of any small numbers is expressed, then we use negative exponents for base 10. For example: 0.0002 = 2 x 10 -4 , where -4 is the negative exponent.

From the above, we can say that the number greater than 1 can be written as the expression with positive exponent, whereas the numbers less than 1 with negative exponent.

Problems and Solutions

Question 1: Convert 0.00000046 into scientific notation.

Solution: Move the decimal point to the right of 0.00000046 up to 7 places.

The decimal point was moved 7 places to the right to form the number 4.6

Since the numbers are less than 10 and the decimal is moved to the right. Hence, we use a negative exponent here.

⇒ 0.00000046 = 4.6 × 10 -7

This is the scientific notation.

Question 2: Convert 301000000 in scientific notation.

Solution: Move the decimal to the left 8 places so it is positioned to the right of the leftmost non zero digits 3.01000000. Remove all the zeroes and multiply the number by 10.

Now the number has become = 3.01.

Since the number is greater than 10 and the decimal is moved to left, therefore, we use here a positive exponent.

Hence, 3.01 × 10 8 is the scientific notation of the number.

Question 3:Convert 1.36 × 10 7 from scientific notation to standard notation.

Solution: Given, 1.36 × 10 7 in scientific notation.

Exponent = 7

Since the exponent is positive we need to move the decimal place 7 places to the right.

1.36 × 10 7 = 1.36 × 10000000 = 1,36,00,000.

Practice Questions

Problem 1: Convert the following numbers into scientific notation.

Problem 2: Convert the following into standard form.

• 2.89 × 10 -6
• 9.8 × 10 -2

Frequently Asked Questions on Scientific Notation – FAQs

How do you write 0.00001 in scientific notation, what are the 5 rules of scientific notation, what are the 3 parts of a scientific notation, how do you write 75 in scientific notation, how do you put scientific notation into standard form.

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Scientific Notation

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Last modified on August 3rd, 2023

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Scientific notation is a special way of representing numbers which are too large or small in a unique way that makes it easier to remember and compare them. They are expressed in the form (a × 10n). Here ‘a’ is the coefficient, and ‘n’ is the power or exponent of the base 10.

The diagram below shows the standard form of writing numbers in scientific notation:

Thus, scientific notation is a floating-point system where numbers are expressed as products consisting of numbers between 1 and 10 multiplied with appropriate power of 10. It helps to represent big and small numbers in a much easier way.

The speed of light(c) measured in a vacuum is approximately 300,000,000 meters per second, which is written as 3 × 10 8 m/s in scientific notation. Again the mass of the sun is written as 1.988 × 10 30 kg. All these values, if written in scientific form, will reduce a lot of space and decrease the chances of errors.

Scientific Notation Rules

We must follow the five rules when writing numbers in scientific notation:

• The base should always be 10
• The exponent (n) must be a non-zero integer, positive or negative
• The absolute value of the coefficient (a) is greater than or equal to 1, but it should be less than 10 (1 ≤ a < 10)
• The coefficient (a) can be positive or negative numbers, including whole numbers and decimal numbers
• The mantissa contains the remaining significant digits of the number

How to Do Scientific Notation with Examples

As we know, in scientific notation, there are two parts:

• Part 1: Consisting of just the digits with the decimal point placed after the first digit
• Part 2: This part follows the first part by × 10 to a power that puts the decimal point where it should be

While writing numbers in scientific notation, we need to figure out how many places we should move the decimal point. The exponent of 10 determines the number of places the decimal point gets shifted to represent the number in long form.

There are two possibilities:

Case 1: With Positive Exponent

When the non-zero digit is followed by a decimal point

For example, if we want to represent 4237.8 in scientific notation, it will be:

• The first part will be 4.2378 (only the digit and the decimal point placed after the first digit)
• The second part following the first part will be × 10 3 (multiplied by 10 having a power of 3)

Case 2: With Negative Exponent

When the decimal point comes first, and the non-zero digit comes next

For example, if we want to represent 0.000082 in scientific notation, it will be:

• The first part will be 8.2 (only the coefficient in decimal form and the decimal point placed after the first digit)
• The second part following the first part will be × 10 -5 (multiplied by 10 having a power of -5)

Here is a table showing some more examples of numbers written in scientific notation:

Let us solve some more word problems involving writing numbers in scientific notation.

Write the number 0.0065 in scientific notation.

0.0065 is written in scientific notation as: 6.5 × 10 -3

Convert 4.5 in scientific notation.

4.5 is written in scientific notation as: 4.5 × 10 0

Write 53010000 in scientific notation.

53010000 is written in scientific notation as: 5.301 ×10 7

Light travels with a speed of 1.86 x 10 5 miles/second. It takes sunlight 4.8 x 10 3 seconds to reach Saturn. Find the approximate distance between Sun and Saturn. Express your answer in scientific notation.

As we know, Distance (d) = Speed (s) × Time (t), here s = 1.86 x 10 5 miles/second, t = 4.8 x 10 3 seconds = 8.928 x 10 8 miles

Other Ways of Writing in Scientific Notation

We sometimes use the ^ symbol instead of power while writing numbers in scientific notation. In such cases, the above number 4237.8, written in scientific notation as 4.2378 × 10 3 , can also be written as 4.2378 × 10^3 Similarly, calculators use the notation 4.2378E; here, E signifies 10 × 10 × 10

• Converting Scientific Notation to Standard Form
• Multiplying Numbers in Scientific Notation
• Dividing Numbers in Scientific Notation
• Adding and Subtracting Numbers in Scientific Notation

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A Complete Guide to Scientific Notation (Standard Form)

Scientific notation: video lesson, what is scientific notation.

Scientific notation can be used to represent both decimals less than one whole and very large whole numbers.

Scientific notation is a concise way to represent both very large or very small numbers. Scientific notation involves a number between 1 and 10 being multiplied to a power of 10. The power of 10 is positive if the number being represented is large and it is negative if the number is small.

• A positive power of 10 tells us how many digits the number has between its first digit and the decimal point
• A negative power of 10 tells us how many zero digits are in front of the non-zero digits.

For example, 6×10 5 is a 6 followed by 5 digits. It represents 600 000.

For example, 6×10 -5 is a 6 with 5 zeros before it. It represents 0.00006.

How Scientific Notation is Used in Real Life

Scientific notation is used whenever very large or very small numbers are involved. For example in physics, astronomy, chemistry and finance.

Some examples of scientific notation used in real-life include:

• The very large distances involved in astronomy. For example, the distance between the Earth and the Sun is said to be 93 million miles, which can be written as 9.3 × 10 7 miles.
• In Physics, very large or very small known constants are represented in scientific notation such as the speed of light (2.998 × 10 8 meters per second) or Planck’s constant (6.63 × 10 -34 joule-seconds).
• In chemistry, the mass of atoms, molecules and molar mass is often written in scientific notation. For example, Avogadro’s number is 6.02 × 10 23 particles per mole.
• In finance and economics, figures such as national debt, gross domestic product (GDP), or market capitalisation are often expressed in scientific notation. For instance, if a country’s GDP is $2.5 trillion, it would be expressed as 2.5 × 10 12 dollars.
• Engineers often work with values that span many orders of magnitude, such as voltage, current, or power.
• In medicine and biology, scientific notation is used to express values like cell counts, DNA base pairs, or concentrations of substances in body fluids.

In science, abbreviations are commonly used to refer to given orders of magnitude such as:

Here are some common values of scientific notation listed on a number line from least to greatest.

Scientific Notation Explained

Scientific notation is used when a very large or very small number is too long to write in the normal manner. In scientific notation numbers are written as a number between 1 and 10 multiplied by a power of 10. For example, 32500 is written as 3.25×10 4 .

Scientific notation is a more concise way to write long numbers whilst indicating their order of magnitude.

Scientific notation is also commonly known as standard form. Both scientific notation and standard form refer to numbers written as a number between 1 and 10 multiplied by 10 to the power of some other positive or negative number.

For example consider the following lists:

• 10 3 = 1000
• 10 4 = 10000

The power that 10 is raised to in each case is equal to the number of digits after the 1.

• 3 × 10 1 = 30
• 3 × 10 2 = 300
• 3 × 10 3 = 3000
• 3 × 10 4 = 30000

The power that 10 is raised to in each case is equal to the number of digits after the 3.

A number written in scientific notation always consists of the following parts:

• A number between the values of 1 and 10
• A multiplication by 10 raised to a power

The name for a number not written in scientific notation is simply ‘standard notation’ or ‘decimal notation’. When comparing a number written in standard notation to a number written in scientific notation, it may also be referred to as ‘expanded form’.

The first step is to write the first part of the number which must always be between 1 and 10.

We take 32500 and write it as 3.25 so that it is now a number that is bigger than 1 and less than 10.

The next step is to find the power of 10 that 3.25 is raised to to make it equal to 32500.

The power that ten is raised to is equal to the number of digits after the first digit.

In 32500, there are 4 digits after the first digit of 3. Therefore the power that 10 is raised to is 4.

32500 can be written as 3.25 × 10 4 .

Scientific notation is often used on calculator displays (since the calculator screen cannot fit very long numbers). On a calculator, scientifi notation is written using ‘E’ to represent × 10^.

For example, 32500 can be written as 3.25 × 10 4 or alternatively as 3.25E4.

In scientific notation, E stands for × 10 raised to the power of the number that comes after the E.

Scientific notation can also be used to write very small numbers by using negative powers of 10.

We can see that the negative power of 10 describes how many zeros are at the start of the decimal number.

For example, 10 -3 has 3 zeros when written as 0.001.

The number 0.000034 can be written as 3.4 × 10 -5 . There are 5 zeros before the digits of 3 and 4.

Rules of Scientific Notation

Numbers written in scientific notation must adhere to the following rules:

• The number before the multiplication sign must always be at least 1 and less than 10.
• There is always a multiplication by a power of 10 (as opposed to a multiplication by any other number).
• The power of 10 can be negative, zero or positive.
• If the number being represented is 10 or more, the power of 10 is positive.
• If the number being represented is 10 or more, the power of 10 is equal to the number of digits that come after the first digit of the number.
• If the number being represented is less than 1, the power of 10 is negative.
• If the number being represented is less than 1, the power of 10 is equal to (-1) multiplied by the number of zeros at the start of the number.
• If the number being represented is greater than 1 but less than 10, the power of 10 is zero.

How to Write Large Numbers in Scientific Notation

• Write a decimal point after the first digit of the number to form a number between 1 and 10.
• Multiply this by a power of 10, where the power is equal to the number of digits after the first digit of the large number.

For example, write 630000 in scientific notation:

Step 1. Write a decimal point after the first digit of the number to form a number between 1 and 10

A decimal point is placed after the 6 to make 6.30000 which is just 6.3

6.3 is a number between 1 and 10.

Step 2. Multiply this by a power of 10, where the power is equal to the number of digits after the first digit of the large number

The first digit of 630000 is 6. We count the number of digits after this first digit of 6.

There are 5 digits after the first digit. That is, the 3, 0, 0, 0 and 0.

Therefore we multiply by 10 5 .

630000 is written in scientific notation as 6.3×10 5 .

How to Write Small Numbers in Scientific Notation

• Write the non-zero digits as a number between 1 and 10.
• Multiply this by a negative power of 10 equal to the number of zeros in front of the first non-zero digit.

For example, write 0.000008 in scientific notation.

Step 1. Write the non-zero digits as a number between 1 and 10

The zeros at the start of 0.000008 are ignored to obtain the non-zero digit of 8.

8 is written because it is a number between 1 and 10.

Step 2. Multiply this by a negative power of 10 equal to the number of zeros in front of the first non-zero digit

The first non-zero digit in 0.000008 is the 8.

The number of zeros in front of this non-zero digit is 6.

Since there are 6 zeros, we multiply the 8 by 10 to the power of -6.

0.000008 is written in scientific notation as 8×10 -6 .

Small numbers less than one whole are written in scientific notation with a negative exponent representing the number of zeros at the start of the number.

For example, write the number 0.0257 in scientific notation.

The zeros at the start of 0.0257 are ignored and the decimal point is placed after the first non-zero digit of 2.

This results in 2.57 which is a number between 1 and 10.

The first non-zero digit is 2.

There are 2 zeros before the 2 and so, we multiply 2.57 by 10 to the power of -2.

0.0257 is written in scientific notation as 2.57×10 -2 .

The power of the 10 is equal to the number of zeros at the start of the number multiplied by -1.

How to Read Scientific Notation

• If the power of 10 is positive, this is the number of digits after the first digit. So 2.11×10 6 has 6 digits after the 2. It is written as 2110000.
• If the power of 10 is negative, this is the number of zeros in front of the digits. So 5.3 × 10 -3 has 3 zeros at the start. It is written as 0.0053.

Scientific Notation for a Large Number

The power of 10 is 6 and so, there are 6 digits after the 2.

We already have two digits of 1 and so, four more 0 digits are needed.

Scientific Notation for a Small Number

The power of 10 is negative 3.

Negative powers tell us how many 0 digits are at the start of the number.

We put three 0 digits and then the 5 and the 3.

The decimal point always comes after the first 0 digit.

For example, 2.93 × 10 4 has 4 digits after the first digit of 2.

Following the 2, there is a 9 then a 3. We need two more 0 digits to obtain 4 digits after the 2.

2.93 × 10 4. = 29300.

For example, 1.04 × 10 -4 has a negative power of 10.

Therefore this power is equivalent to the number of zeros at the start of the number. There will be 4 zeros followed by the digits of 1, 0 and 4.

1.04 × 10 -4 = 0.000104.

Notice that the power of 10 is only equal to the number of zeros at the start of the number, not the total number of zeros in the answer.

Examples of Scientific Notation

Notice that as an extra zero digit is added to the value, the exponent in scientific notation increases by 1.

Notice that as we move from one thousand to one million to one billion to one trillion, three zero digits are added and so, the exponent in scientific notation increases by 3.

Here are some common values listed in scientific notation:

The power of zero in 1×10 0 means that zero tens have been multiplied by.

That is, 10 0 = 1 and so 1×10 0 simply means 1×1 which is just equal to 1.

Here are some examples of numbers written in scientific notation:

How to Round Numbers in Scientific Notation

It is common for numbers to be written to a particular number of significant figures and then written in scientific notation. This is because numbers in scientific notation are very large or very small and so, the final digits do not really impact on the overall size of the number.

Typically, numbers in scientific notation are given to 2 or 3 decimal places although further accuracy may be required.

It is common to round numbers in scientific notation to 3 significant figures.

For example, write 125364 in scientific notation, rounded to 3 significant figures.

The first step is to round the number to 3 significant figures.

That is, we look at the first 3 non-zero digits of the number, which are 1, 2 and 5. We then look at the next digit after this to decide if the 3rd significant figure of 5 remains as a 5 or rounds up to a 6.

Only round up the 3rd significant figure if the 4th significant figure is equal to 5 or more.

Since the digit after the 5 is a 3, we do not round up.

We write 125 and then replace the other digits with zeros, so that 125364 written to 3 significant figures is 125000.

We now write this in scientific notation as 1.25 × 10 5 , since there are 5 digits after the first digit of 1 in the number 125000.

For example, write the number 0.06627 in scientific notation to 3 significant figures.

Significant figures are counted after the zeros at the start of the number.

We look at the first 3 significant figures of the number which are 6, 6 and 2.

We only round up the 3rd significant figure if the 4th significant figure is equal to 5 or more.

The 4th significant figure is a 7 and so, we round the 2 up to a 3.

0.06627 rounded to 3 significant figures is 0.0663.

This number starts with 2 zeros, so writing it in scientific notation we have 6.33 × 10 -2 .

How to Add and Subtract Numbers in Scientific Notation

The ease of adding and subtracting numbers in scientific notation depends on whether the size of the exponents are the same in each number.

To add or subtract numbers in scientific notation that have the same exponent, simply add or subtract the coefficients and keep the exponent the same. For example, 3×10 5 + 4 × 10 5 is (3+4) × 10 5 which equals 7 × 10 5 .

In this example, the exponents on both 3×10 5 and 4×10 5 are both 5.

Therefore, we can simply add the 3 and the 4 together to obtain 7, whilst the exponent in the answer remains as 5.

3×10 5 + 4×10 5 = 7×10 5 .

This addition is essentially the same as 300 000 + 400 000 = 700 000.

For example, 3×10 -4 – 1×10 -4 = (3-1)×10 -4 which is just 2×10 -4 .

In this example, we simply subtract the coefficients and keep the exponent as -4.

To add or subtract numbers in scientific notation that have different exponents, it is easiest to convert the numbers to standard notation first. Then perform the addition or subtraction and write the result back in scientific notation if needed.

In the example of 3.21×10 3 + 2×10 4 , the exponents are different sizes.

• Converting 3.21×10 3 to standard notation, we have 3 210. The exponent of 3 means there are 3 digits after the first digit.

After the 3, we have the 2 then the 1 and so, another digit of 0 is needed to make 3 digits after the first digit in total.

Converting 2×10 4 to standard notation, we have 20 000. That is, we have 4 digits after the first digit of 2.

Now, these numbers can be added following the standard addition process.

3 210 + 20 000 = 23 210.

We can write this in scientific notation if needed as 2.321×10 4 since there are 4 digits after the first digit in 23 210.

Here is an alternate method for adding numbers in scientific notation when they have different exponents.

The same numbers as above are used but here we are demonstrating a different method to solve the problem.

To add or subtract numbers in scientific notation, first convert the numbers to have the same exponent. To do this, multiply the coefficient of the number with the largest exponent by 10 each time the exponent is reduced by 1.

For example, in 3.21×10 3 + 2×10 4 , we wish to reduce the exponent in 2×10 4 from a 4 to a 3 so that it is the same exponent as in 3.21×10 3 .

Since we need to reduce the exponent by 1, we multiply the 2 by 10 to make 20.

2×10 4 is the same as 20×10 3 .

Therefore 3.21×10 3 + 2×10 4 is rewritten as 3.21×10 3 + 20×10 3 .

Now the coefficients can be added so that 3.21×10 3 + 20×10 3 = (3.21+20)×10 3 .

This equals 23.21×10 3 which can be readjusted to 2.321×10 4 as we divide the coefficient by 10 as we increase the exponent of 10 by 1.

Here is an example of subtracting numbers in scientific notation by first converting them to have the same exponents.

In the example of 5.3×10 5 – 7.9×10 4 , we need to change the exponent of 5.3×10 5 to a 4.

To do this, we multiply the 5.3 by 10 to make 53.

5.3×10 5 is the same as 53×10 4 .

Therefore 5.3×10 5 – 7.9×10 4 can be written as 53×10 4 – 7.9×10 4 .

Now the numbers have the same exponent, subtract the coefficients like so:

(53 – 7.9)×10 4 = 45.1×10 4 .

Finally, this is rewritten so that the coefficient is a number between 1 and 10. We divide 45.1 by 10 and increase the exponent by 1 to compensate.

45.1×10 4 = 4.51×10 5 .

How to Multiply and Divide Numbers in Scientific Notation

To multiply numbers in scientific notation, multiply the coefficients and add the exponents. For example, (2×10 3 ) × (3×10 5 ) = 6×10 8 . The coefficients of 2 and 3 were multiplied to make 6 and the exponents of 3 and 5 were added to make 8.

This occurs since exponents are always added when they are multiplied together.

Here is an example of multiplying numbers in scientific notation: (4.5×10 9 ) × (5.2×10 -2 ).

Firstly, the coefficients of 4.5 and 5.2 are multiplied to obtain 23.4.

Secondly, the exponents are added so that 9 + -2 = 7.

Therefore, (4.5×10 9 ) × (5.2×10 -2 ) = 23.4×10 7 .

However, 23.4 must be written as a number between 1 and 10.

We divide 23.4 by 10 to obtain 2.34 and we increase the exponent from 7 to 8 to compensate.

(4.5×10 9 ) × (5.2×10 -2 ) = 2.34×10 8 .

To divide numbers in scientific notation, divide the coefficients and subtract the exponents. For example, (6×10 7 ) ÷ (2×10 2 ) = 3×10 5 . The coefficients of 6 and 2 were divided to obtain 3 and the exponent of 2 was subtracted from the exponent of 7 to obtain 5.

This occurs since exponents are subtracted when they a division takes place.

Here is another example of dividing numbers in scientific notation: (8.4×10 4 ) ÷ (2.5×10 -3 ).

Firstly, the coefficients are divided so 8.4 ÷ 2.5 = 3.36.

Secondly, the exponents are subtracted. 4 – -3 is the same as 4 + 3 which equals 7.

(8.4×10 4 ) ÷ (2.5×10 -3 ) = 3.36×10 7 .

Scientific Notation on Calculator

Scientific notation is commonly used in calculator displays.

To write scientific notation on a calculator, use the button labelled as ‘×10 𝑥 ‘ or just 10 𝑥 ‘.

• Enter the coefficient (the number between 1 and 10)
• Then press the ‘×10 𝑥 ‘ button
• Then enter the power of 10

Here are the instructions for entering scientific notation on some common calculators:

Casio Fx-300

The ‘×10 𝑥 ‘ button is found in the middle of the bottom row.

Casio Fx-82

Casio fx-cg50.

The ’10 𝑥 ‘ option is found above the ‘log’ button. Press ‘2nd’ then ‘log’ to write numbers in scientific notation.

• Press the ‘2nd’ button
• Then press the ‘log’ button to select the ’10 𝑥 ‘ option

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1.1: Scientific Notation - Writing Large and Small Numbers

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Learning Objectives

• Express a large number or a small number in scientific notation.
• Carry out arithmetical operations and express the final answer in scientific notation

Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes. In scientific notation, these numbers are expressed in the form

\[ N \times 10^n \nonumber \]

where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (10 0 = 1). The number 10 is called the base because it is this number that is raised to the power \(n\). Although a base number may have values other than 10, the base number in scientific notation is always 10.

A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows:

• If the decimal point is moved to the left n places, n is positive.
• If the decimal point is moved to the right n places, n is negative.

Another way to remember this is to recognize that as the number N decreases in magnitude, the exponent increases and vice versa. The application of this rule is illustrated in Example \(\PageIndex{1}\).

Example \(\PageIndex{1}\): Expressing Numbers in Scientific Notation

Convert each number to scientific notation.

Addition and Subtraction

Before numbers expressed in scientific notation can be added or subtracted, they must be converted to a form in which all the exponents have the same value. The appropriate operation is then carried out on the values of N. Example \(\PageIndex{2}\) illustrates how to do this.

Example \(\PageIndex{2}\): Expressing Sums and Differences in Scientific Notation

Carry out the appropriate operation and then express the answer in scientific notation.

• \( (1.36 \times 10^2) + (4.73 \times 10^3) \nonumber\)
• \((6.923 \times 10^{−3}) − (8.756 \times 10^{−4}) \nonumber\)

Multiplication and Division

When multiplying numbers expressed in scientific notation, we multiply the values of \(N\) and add together the values of \(n\). Conversely, when dividing, we divide \(N\) in the dividend (the number being divided) by \(N\) in the divisor (the number by which we are dividing) and then subtract n in the divisor from n in the dividend. In contrast to addition and subtraction, the exponents do not have to be the same in multiplication and division. Examples of problems involving multiplication and division are shown in Example \(\PageIndex{3}\).

Example \(\PageIndex{3}\): Expressing Products and Quotients in Scientific Notation

Perform the appropriate operation and express your answer in scientific notation.

• \( (6.022 \times 10^{23})(6.42 \times 10^{−2}) \nonumber\)
• \( \dfrac{ 1.67 \times 10^{-24} }{ 9.12 \times 10 ^{-28} } \nonumber \)
• \( \dfrac{ (6.63 \times 10^{−34})(6.0 \times 10) }{ 8.52 \times 10^{−2}} \nonumber \)

Helping with Math

Scientific Notation

Math is used by scientists at all levels to explain their thoughts. A physicist must frequently employ extremely small or very big numbers when explaining natural parameters. Some figures are so small that they are difficult to express using standard notation. The mass of an electron, for example, is 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 005

Some numbers are so big that traditional units such as millions, billions, and trillions cannot be used to describe them. The mass of the Sun, for example, is believed to be 8,000,000,000,000,000,000,000,000,000,000 kg, or 8 followed by 30 zeros. It can be difficult to write such high numbers. Engineers and physical scientists implement scientific notation to describe these values efficiently and simply. Hence we can define scientific notation as follows

A number is expressed in scientific notation as a product of any integer between 1 and 10 to the 10th power. Because it shortens the notation, scientific notation is most utilized when dealing with huge quantities or numbers with numerous digits.

General form of scientific notation

Scientific notation has a general representation of a x 10 b (where “b” is an integer and “a” is any real value between 1 and 10). Only include significant figures in the real number, “a,” when writing in scientific notation.

The coefficient , base, and exponent are the three main components of scientific notation.

Examples of scientific Notation

960,000,000 can be expressed as 9.6 x 10 8 in scientific notation.    a = 9.6    b = 8   (according to general form)

0015000 can be written as 1.5 x 10 4     Here a = 1.5     b = 8 

Some more examples of scientific notation 

0.00000045   is written in scientific notation as 4.5 x 10 -7

675,000,000,000    can be written as 6.75 x 10 11

00583000   can be written as  5.83 x 10 5

0.000005000       can be written as 5x 10 6     OR 0.5 x 10 5

So, in these examples we analyzed that value of ‘’a’’ cannot be greater than 10 and b can be any integer without distinguish of positive or negative integer.

Rules for scientific notations

We must use the following rule to determine the power or exponent of ten :

• The starting point should always be ten.
• The exponent has to be a non-zero integer, which might be positive or negative .
• The coefficient’s magnitude is more than or equal to one, but it should be less than ten.
• Positive and negative numbers, as well as whole and decimal numbers , can be used as coefficients.
• The remainders of the number’s significant digits are carried by the mantissa.
• With the help of the representation below, we can see how many places we need to shift the decimal point after the single-digit value.
• If the given integer is a multiple of 10, the digits must be shifted to the left, and the power of ten will be positive.

480000000 = 4.8 × 10^8

4230000000 = 4.23 × 10^9              

60500000 = 6.05 x 10^7             

0.000000098 = 9.7 x 10^-8

0.0000212 = 2.12 x 10^-5 are some instances of scientific notation: 480000000 = 4.8×10^8

1230000000 = 1.23 × 10^9                          

50500000 = 5.05 x 10^7               

0.000000097 = 9.7 x 10-8 0.0000312 = 3.12 x 10-5

How to write Scientific notation ?

The large or very small number written in ordinary form is called the standard form of the number.

We can convert standard form to scientific notation and scientific notation to standard form by following simple steps.

First we see the steps for conversion of standard form to scientific notation.

 Shift the decimal point to the left till you have a value that is higher than 1 but less than 10.

Step 2: 

Look at the number of decimal places to the left of the decimal point and use that number to calculate the positive power of ten.

Multiply the decimal (from Step 1) by a factor of ten (in Step 2).

For a value of less than 1

 Shift the decimal point to the right until you can get a value that is more than or equal to

Or less than 10 and equal to 1

 Check the amount of decimal places the decimal point has been shifted to Correct, then utilize that amount as a negative power of ten

Step 3: 

 For example 

0.00025 = 2.5 × 10^-4

A standard form number less than 1 has negative coefficient.

Conversion of scientific notation to standard form

Let’s see if we can return our new scientific notation numbers to their standard form.

Start the process with the number (not the power of ten part).

For each power of 10 you have, move the decimal point one place. Move the decimal point to the right if the powers of ten are positive. Move the decimal point one place to the left if they’re negative.

 When we’re out of digits, fill in with 0s.

If your number is large, ensure sure the decimal places to the left of the decimal point are divided by commas into three groups of three.

For example 

Scientific notation 4.45 × 10^2

1. Take the number and drop the ten and its exponent, as well as the multiplication sign (power)       4.45

2.  Because the power of ten was positive, move the decimal one point to the right for each power of ten. Hence 

3.  Because there are only three digits, no commas are required.

Some example for conversion of scientific to general

1.45 × 10^-5       = 0.0000145

3.56 ×10^7     = 35,600,000

1.8945 ×10^10     = 18,945,000,000

Arithmetic in Scientific notation

Numbers expressed in scientific notation can be added, subtracted, multiplied, and divided while remaining in scientific notation.

The processes for adding and subtracting two numbers in scientific notation are listed below.

By shifting the decimal point of its decimal number, rewrite the number with the smaller exponent to have the same exponent as the number with the larger exponent.

Decimals should be added or subtracted. The power of ten is unchangeable.

If needed, convert your result to scientific notation.

Here’s an illustration.

(5.8 × 10^4) + (4.12 × 10^5)

First, observe the numbers 4 and 5 in the exponents. You’ll need to rewrite 5.8 × 10^4 to make it have a 5 exponent. Because the exponent must be increased by one, the decimal point will be moved one place to the left.

5.8 ×10^5 replace 0. 57 × 10^4

Re – write this problem now.

(0.58 × 10^5)+(4.12  ×10^5)

Then, to use what we’ve learned about decimal addition, add the decimal numbers together the power of ten remains constant.

=> 0.58+4.12 

= 4.70 is the coefficient part of the scientific notation

• 4.70 × 10^5 is the required answer.

Similar for subtracting the numbers in scientific notation

(1.23 × 10^6) – (1.20 × 10^6)

              =1.23-1.20

:: exponent is same so it is preserved 

• 0.03 × 10 ^6 is the required answer after subtraction

The processes for multiplication or division numbers in scientific notation are outlined below.

Divide or multiply the decimal numbers.

Add/subtract the exponents of the powers of ten to multiply/divide them.

If necessary, convert your solution to scientific notation.

Here’s an example of two decimal integers being multiplied.

Multiply (3.4 × 10^2) by 6.2 × 10^6.

First, use what you’ve learned about decimal multiplication to multiply the decimal numbers .

3.4×6.2=21.08

After that, add the exponents of the powers of ten.

10^2 × 10^6=10^(2+6) = 10^4

Combine the results now.

(3.4 × 10^2).(6.2 × 10^6)==(3.4 × 6.2) × (10^2 × 10^6) = 21.08 × 104

Convert your solution to scientific notation at the end. You must rewrite 21.08 × 10^8 in such a way that the decimal value is at least 1 but not more than 10. Move the decimal point to the left one space. Increase the exponent on the 10 by one to keep the overall value the same.

21.08 × 10^8 becomes 2.108 × 10^9

Here’s an illustration of how to divide two decimal integers.

Divide (8.4 × 10^5)(1.4 × 10^2) 

First, use what we have learned about 8.4 ÷ 1.4= 6 to divide the decimal values.

Subtract the exponents from the powers of ten to divide them. It’s important to keep in mind that subtracting a negative number is just the same as addition the positive version.

10^5 ÷ 10^2 = 10^(5−2)  = 10^3

(8.4 × 10^5) ÷ (1.4 × 10^2)  = (8.4 ÷ 1.4) × (10^5 ÷ 10^2)

Last but not least, double-check that your answer is written in scientific notation. Because 6 is less than 10 but greater than 1, your answer is in scientific notation.

The solution is (8.4 × 10^ 5) ÷ (1.4 × 10^2) = 6.0 × 10^7.

How to write scientific notations in calculator

Take, for instance, the value 1.52 x 105 on a TI-30 calculator. To type this number in scientific notation, start by typing 1.52 into the calculator. After then, press the [2nd] key, followed by the x^-1 key, which has EE displayed above it. “x 10 to the power of” is what EE stands for. Type the exponent 5 after you’ve pressed the [EE] key. The display should look like this: 1.52 105 **, with a 05 in smaller print in the upper right corner of the display. The [EE] key is not available on the TI-30XS Multi-View TM scientific calculators. Instead, they offer a shortcut key, [x10n], which can be used to enter scientific notation exponents.

For a TI-83/84 calculator

Click the mode button and select SCI on the top line to keep the number you input remains in scientific notation. When you press the enter key when in NORMAL mode, the number will be extended.

Let’s utilize the same 1.52 x 10 ^5 example as the TI-30. 1.52 should be typed in. Then press the [2nd] key, followed by the [ ] “comma” key (it’s the key with the EE above it). After that, enter the exponent. It will be written as 1.52E05, which stands for 1.52 x 10 ^5

Why we use scientific notation

When we’re interested in a career in math, engineering, or science (or already work in one of these professions). We’ll almost certainly need to use scientific notation. Computer scientists and astronomy, in particular, rely on scientific notation on a daily basis since they work with microscopic particles all the way up to gigantic celestial objects and require a system that can manage such a wide range of numbers.

One of the benefits of scientific notation is that it helps you to be more precise with your numbers, which is important in those fields. Rather than rounding to a figure that is simpler to say or write, scientific notation allows you to be extremely precise with your numbers without making them unmanageable.

Writing scientific papers

In writing a scientific study article, you may need to utilize scientific notation because scientific investigations can contain very large or very small figures that must be precise. Consider the following scenario: If you’re dealing with the mass of particles or lengths in the universe, you don’t want to see pages full of numbers with digit after digit or numbers with seemingly endless zeroes! You also don’t want to round up or down too much, as this could skew your results and undermine your trustworthiness.

Use of scientific notation in science and engineering

Scientists and engineers frequently collaborate with both large and small groups of people. In this case, the standard practice of utilizing commas and leading zeroes proves to be extremely inconvenient. Scientific notation is a technique of representation that is more compact and less prone to errors. There are two parts to the number: an accuracy part (the mantissa) and a scale part (the significance) (the exponent, being a power of ten). 23,000, for example, may be written as 23 times 10 to the third power (that is, times one thousand). The exponent can be thought of as the lot of locations to the left of the decimal point. Because writing “times 10 to the X power” is inconvenient, a shorthand approach is utilized, in which the letter E replaces “times 10 to the X power” (which stands for exponent). As a result, 23,000 can be expressed as 23E3. 45E9 represents the value of 45,000,000,000.

It’s worth noting that this figure might alternatively be written as 4.5E10 or perhaps even 0.45E11. The only distinction between scientific and engineering notation is that the exponent in engineering notation is always a multiple of three. As a result, 45E9 is correct engineering notation, however 4.5E10 is not. E is commonly denoted by a “EE” or “EXP” button on most scientific calculators. Depressing the keys 4 5 EE 9 would be the procedure of entering the value 45E9.

The exponent is negative for fractional values and can be thought of as the number of places the decimal point must be pushed to the right. 0.00067 can so be written as 0.67E3, 6.7E4, or even 670E6. Only the first also last of these three can be used as engineering notation.

Engineering notation takes a step further by replacing the multiples of three for the exponent with a set of prefixes. The prefixed are as follows:

10^6   =   E6 = Mega (M)

10^3 = E3 = Kilo (k)

10^9 = E9 = Giga (G)

10^12 = E12 = Tera (T)

10^-6 = E-6 = Micro () and so forth

23,000 volts, for example, can be expressed as 23E3 volts or simply 23 kilovolts.

This writing is much simpler than the standard form for processing a large range of numbers, in addition to being more concise. Simply multiply the precise parts and add the exponents when multiplication. Divide the accuracy portions and eliminate the exponents when dividing. For example, multiplying 23,000 by 0.000003 may appear to be a difficult process. This is 23E3 times 3E6 in technical notation. 69E3 is the outcome (that is, 0.069). It will become second nature after enough experience that kilo (E3) multiplied by micro (E6) equals milli (E3). This will make lab calculations a lot easier. 42,000,000 divided by 0.002 equals 42E6 divided by 2E3, or 21E9 (the exponent is 6 minus a negative 3, or 9)

Before adding different, make absolutely sure the exponents are equal (scaling if necessary) before adding or subtracting the precision sections. For instance, 2E3 plus 5E3 equals 7E3. 2E3 plus 5E6 is like 2E3 plus 5000E3, or 5002E3 by comparison (or 5.002E6).

Overall, scientific notation is a handy technique to write and work with extremely large or extremely small numbers. Scientific notation is useful for persons undertaking academic and professional work in math and science, even if it may seem difficult to conceive applying it in everyday life. A scientific notation calculator and converter make using this shorthand more easily for anyone studying or working in these subjects.

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1.2: Scientific Notation

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Sometimes we need express very large or very small numbers. For instance, how many cells are there in the human body? And how large, in inches, is a cell? The answers to questions like this can be difficult to understand when expressed in regular notation:

Number of cells in the human body \( = 37,200,000,000,000\)

Length of a cell \( = 0.0011811\) inches

For numbers like this, it can be helpful to express them in scientific notation.

In this section, you will learn to:

• Recognize and explain scientific notation
• Use scientific notation to compare numbers

Basics of Scientific Notation

The goal of scientific notation is to make numbers shorter to write and easier to compare. It uses two tools you saw in the previous chapter: the base \(10\) system, and exponents. Let's start with the definition and then see how it is used in a few examples.

Definition: Scientific Notation

A number is written in scientific notation when it is expressed as a number between \(1\) and \(10\) that is multiplied by a power of 10. As an example, we could write

\[37,200,000,000,000 = 3.72 \times 10^{13}\]

The exponent on the 10 tells you how many places the decimal point has been moved. In this case, the exponent was moved 13 places to the left.

Scientific notation leverages the base \(10\) system to write very large or very small numbers in a compact way. In the definition above, we see that a positive exponent moves the decimal point to the left. Think for a moment about why this makes sense, based on the last section. Then try the following example.

Example \(\PageIndex{1}\)

Explain why the numbers \(37,200,000,000,000\)  and \(3.72 \times 10^{13}\) are equal. 

Remember from the previous section that exponents represent repeated multiplication, and we can tell a power of \(10\) by the number of zeroes has following the starting \(1\). Therefore,

\[10^{13} = 10,000,000,000,000 \]

When we multiply this number by \(3.72\), we can see (using either long-hand multiplication, or by thinking about the base 10 expansion), that

\[3.72 \times 10,000,000,000,000 = 37,200,000,000,000\]

Therefore, \(37,200,000,000,000 = 3.72 \times 10^{13}\).

The ideas above will allow us to write any number in scientific notation. The only difference for very small numbers -- those expressed as long decimals -- is that we use negative exponents instead of positive exponents. The following table summarizes how to write both very large and very small numbers in scientific notation.

Let's practice with one more example.

Example \(\PageIndex{2}\)

Convert the number \(.0000049\) to scientific notation.

We see that this is a very small number, between \(0\) and \(1\). This means that we need to use a negative exponent to move the decimal point to the right. We need to move the decimal point from its current location to the place between the \(4\) and the \(9\), so that the resulting number will be between \(1\) and \(10\). If we count places from the current location to the new location, we see that it has to move 6 places. Therefore, \[.0000049 = 4. \times 10^{-6} \]

The process of converting from scientific notation to ordinary notation consists of simply reversing the direction that the decimal point travels, and including an appropriate number of zeroes. Let's see two examples of this. Before looking at the solution, see if you can work backwards to figure it out yourself!

Example \(\PageIndex{3}\)

Convert each of the following to ordinary notation.

• \(9.362 \times 10^{10}\)
• \(5.7 \times 10^{-4}\)
• There is a positive exponent here, which means we expect a very large number to result. We will move the decimal point 10 places to the right, since we are reversing the process of conversion. We will start putting zeroes once we run out of numbers in \(9.362\). Therefore, \[9.362 \times 10^{10} = 93,620,000,000\]
• Here, we have a negative exponent, which means that our result will be a very small number. Since we are reversing the process, we'll move the decimal point 4 places to the left, adding zeroes as we go. Therefore\[5.7 \times 10^{-4} = .00057\]

What do "very large" and "very small" mean? As is often the case with topics in this book, the answer depends on the context in which you are using the number. The choice to write \(453,000,000\)  versus \(4.53 \times 10^8\) is up to whomever is writing that number. In general, they should consider the purpose of the number they are writing and who is likely to read it. In the context of a news article, one might see \(453,000,000\), or perhaps "453 million." The second version of the number gives a sort of alternative scientific notation, in which we've written \(453,000,000 = 453 \times 10^6 = 453 \times 1,000,000\). The previous equation does not give the standard scientific notation — which requires that the number in front be between \(1\) and \(10\) — but it can be a useful way to express numbers verbally!

However, in a scholarly context, scientific notation is often used in papers and studies to express a variety of numbers, especially when there are multiple numbers of a similar type being compared. The following section shows how scientific notation can be used to compare numbers.

Comparing numbers using scientific notation

One of the benefits of scientific notation is that it allows us to compare the relative size of quantities that are very large or very small. This is especially clear when the quantities are different orders of magnitude; that is, when they have different exponents in scientific notation. Using scientific notation can help make this more obvious than ordinary notation.

Example \(\PageIndex{4}\)

Which is larger? \(89000000\) or \(9800000\)?

The most likely way to approach a problem written in this way is to squint at the numbers really hard, and count how many zeroes there are either with your eyes or a tool like a pencil. If you do that, you'll see that there are \(6\) zeroes in the first number and only \(5\) in the second number, so the first number must be larger. However, if we had been given these numbers in scientific notation: \[8.9 \times 10^{7} \text{ or } 9.8 \times 10^{6}\] then it is very clear that the first number is larger, since it is multiplied by a larger power of \(10\). As a side note, these would be considered quantities with different orders of magnitude; \(89000000\) is one order of magnitude larger than \(9800000\).

How might this be useful in context? Consider the following example.

Example \(\PageIndex{5}\)

Many chemicals are harmful if ingested, inhaled, or otherwise consumed by humans. Often, we use blood concentrations of chemicals to define the quantity of a substance that is considered harmful. Mercury and lead are two chemicals known to be harmful at very low levels. Lead can be shown to cause harm at concentrations greater than \(3.5 \times 10^{-6}\) grams per deciliter of blood, whereas mercury can be shown to cause harm at concentrations greater than \(1 \times 10^{-7}\) grams per deciliter of blood.

Use the information above to compare the effects of equal amounts of lead and mercury exposure.

There are many ways to answer this question, of course. The point of this example, and of similar exercises you will find throughout the text, is to practice reading, analyzing, and describing quantitative information in context.

What we see in the paragraph above is that while lead and mercury are both dangerous to humans, even at small quantities, a small amount of mercury is more harmful than the same amount of lead, since the concentration required for harm is lower than that of lead.

For example, someone with \(1.5 \times 10^{-7}\) grams of mercury per deciliter of blood would be above the level of toxicity for mercury, but someone with the same amount of lead — \(1.5 \times 10^{-7}\) grams of lead per deciliter of blood — would be below the level of toxicity for lead, since \(1.5 \times 10^{-7}\) is less than \(3.5 \times 10^{-6}\).

The use of scientific notation allows us to compare these two numbers more clearly than if they were written in standard notation.

And of course, please note: both mercury and lead, at any level, can be very dangerous, and represent serious environmental hazards for many. Please avoid these chemicals as much as possible.

A note about notation

The expression of scientific notation above is common in published textbooks and journals in a variety of fields. However, technology such as spreadsheets, calculators, and computation programs use various shorthand for scientific notation. There are many variations on this as various manufacturers have adopted different conventions, but a fairly common notation is the use of "E" to indicate the power of \(10\) in an expression in scientific notation. For example, your computer might display the number \(587,000,000\) as "5.87E8," which is shorthand for \(5.87 \times 10^8\). Be aware of this when you are using technology to handle very large or very small numbers!

• Convert \(792,000,000,000\)  to scientific notation.
• Convert \(.0000508\)  to scientific notation.
• Convert \(8.6 \times 10^7\) to ordinary notation.
• Convert \(7.2 \times 10^{-9}\) to ordinary notation.
• Which is larger: \(3.02 \times 10^5\) or \(4.02 \times 10^4\)? Explain how you know.
• Which is larger: \(6.11 \times 10^{-15}\) or \(1.001 \times 10^{-13}\)? Explain how you know.
• Write the number in ordinary notation. 
• Write the number in scientific notation. 
• Of the types of notation — including how the number was originally presented in the article — write a 1-2 sentence reflection on which type of notation you believe most clearly expresses the size of the number and why. 
• Include a link to your news article. 

Scientific Writing with Markdown

Introduction.

Markdown is a lightweight markup language with plain text formatting syntax. This article explains how to use Markdown for writing scientific, technical, and academic documents that require equations, citations, code blocks, Unicode characters, and embedded vector graphics. Markdown offers the easiest and most versatile syntax and tools for creating these types of documents.

Markdown was initially designed for creating content for websites (HTML), but we can also create other document formats such as PDF and EPUB using converters like Pandoc . We can also use Markdown to write LaTeX documents more easily compared to using pure LaTeX. Since Markdown files are text files, they need to be converted to a separate output document, unlike MS Word or Google Docs , which displays the output document while editing it. We include software recommendations for writing and converting Markdown.

To help getting started, check out the Markdown-Templates GitHub repository, which demonstrates how to create documents in practice. We explore its contents during this article.

Markdown is not the best choice for documents requiring lots of small customizations in styles, fonts, colors, or outlooks. On the other hand, Markdown excels at creating documents that need little customization or have premade styles or templates available.

We need a converter to convert Markdown to other document formats. Pandoc is the primary tool that we use for converting Markdown into other formats. LaTeX is a typesetting system designed for the production of technical and scientific documentation. Pandoc requires LaTeX installation for creating PDF documents. We can install them from their respective websites, Pandoc and LaTeX .

For creating a static website to write scientific content using Markdown, we recommend using Hugo with Wowchemy . Their respective websites document them extensively, and we recommend to read them for more information.

For technical documentation, it is best to use documenting software recommended for the programming language. For example, we can use Documenter.jl to create documentation using Markdown for Julia projects.

We need an editor to write Markdown effectively. The best options for writing Markdown on a desktop are Atom and Visual Studio Code . A desktop editor should have the following features:

• Support and highlighting for Markdown syntax.
• Live preview for Markdown documents.
• Integrated terminal to run commands for creating documents.
• Ability to input Unicode characters. Unicode characters make writing equations easier. We can search all Unicode characters from the Unicode table . Unicode-math-symbols table contains the mappings between corresponding Unicode characters and LaTeX commands.
• Integrated PDF document viewer that refreshes the view if we create a new document. Alternatively, we can use an external PDF viewer.

For collaborative scientific writing, we recommend Authorea . Authorea is a modern web platform for scientific writing and publishing. It supports Markdown syntax, among many other modern features.

We can also write Markdown on the browser using StackEdit . It integrates with many cloud services such as Google Drive , Dropbox and GitHub . We recommend trying different editors and choosing the one that works best.

To use Atom, install it from their website and then install the following packages for writing Markdown by navigating to Edit > Preferences > Packages : language-markdown , markdown-preview-plus (disable markdown-preview ), platformio-ide-terminal , pdf-view , and latex-completions .

Visual Studio Code

To use Visual Studio Code, install it from their website and then install the following extensions for writing Markdown by navigating to File > Preferences > Extensions : Markdown All in One , Markdown Preview Enhanced , and Unicode Latex . Visual Studio Code comes with an integrated terminal. Now we are ready to start creating Markdown documents.

Creating Documents

The example documents in Markdown Templates are structured as follows:

The Markdown file <filename>.md is where we write the content of the document. We use the bibliography.bib file to store bibliographical entries in BibTeX format, which we can refer to in the Markdown document.

The Makefile contains commands for converting the Markdown file into the desired document format using Pandoc. Pandoc creates the output files to the build/ directory, which Makefile automatically creates if it does not exist.

We define the build directory and the filename at the beginning of Makefile as follows.

Then, we define the command to create the document.

The option --from=markdown tells that input file is a Markdown file. Markdown extensions +tex_math_single_backslash and +tex_math_dollars enable Pandoc to parse equations.

Pandoc-citeproc enables us to use citations in Markdown. Pandoc installation includes it by default. We need to enable it by using the option --filter pandoc-citeproc .

We can execute the Makefile command in the terminal as follows.

Next, we define concrete examples of Makefile for creating PDF, HTML, and EPUB documents.

Markdown extension +raw_tex enables us to insert raw LaTeX inside the Markdown document, and the --pdf-engine=xelatex option enables us to use Unicode characters within the Markdown document.

The --mathjax flag enables math rendering for HTML via Mathjax, and the --self-contained flag includes style sheets to the output document.

For e-books, we need to enable the table of contents needs using --toc flag. We can also include a cover image of size 1600 x 2400 pixels in JPG or PNG formats using --epub-cover-image=<cover-image> option.

Front Matter

We can include document-specific metadata and functionality for the converter in the Front Matter at the top of a <filename>.md file. We write the front matter in YAML between two triple-minus lines --- as follows.

The title , date , and author variables specify information for creating the title. The bibliography variable specifies the location of the bibliography file, link-citations toggles links to citations on and off, urlcolor defines the link color, and the csl variable defines the Citation Style Language . We can find examples of citation styles from Zotero styles and use them by either downloading them or referring directly to the URL of the raw CLS file in citation styles repository.

We can change the layout of a PDF document by including a LaTeX preamble using a Pandoc command or the Front matter. For example, if we want to create an ebook friendly PDF output, we can use the layout described in “Effort to make latex ebook friendly” as follows.

We can include --include-in-header=<layout>.tex option in the Makefile with <layout>.tex file:

Alternatively, we can include the header-includes variable in the front matter:

Basic Syntax

John Gruber’s original spec and Markdown Cheatsheet in GitHub demonstrate the basic Markdown syntax. We recommend reading at least them to understand the basics. In addition to Markdown understanding, basics on HTML can be useful for creating web content if using inline HTML.

Code Blocks

Regular Markdown supports code blocks but does not highlight their syntax. However, converters such as Pandoc will add syntax highlighting for code block as long as we supply the appropriate language for the code block. For example:

Displays as:

We can write inline equations using single dollars $...$ or backslashed parentheses \(...\) and display using double dollars $$...$$ or backslashed square brackets \[...\] . The use of dollar symbols is more common. Optionally, we can add tags \tag{<tag>} for numbering equations and labels \label{<label>} for referring to equations later in text using \ref{<label>} . For example, we can write Cauchy’s integral formula as

Mathjax displays the equation as

We can now refer to the equation using syntax (\ref{1}) which displays as (\ref{1}). Markdown displays inline equations such as $a^2+b^2=c^2$ in the same line as the text, $a^2 + b^2 = c^2.$

Some markdown parsers do not detect the equation mode for characters that are part of the Markdown syntax, which might interfere with parsing. For example, instead of using the asterisk symbol * inside the equation mode, we can use the backslashed ASCII \* , latex command \ast , or the Unicode version ∗ .

Unicode Symbols for Equations

We write equations easier by using Unicode characters for mathematical symbols. For example, instead of \mathbf{x}\in\mathbb{R}^2 we can write $𝐱∈ℝ^2$ for the same output $𝐱∈ℝ^2$. You can input Unicode symbols with editor plugins mentioned in the Editors section.

Colored Equations

Better Explained in their article Colorized Math Equations shows how to use colors to improve the way we can describe equations. The article has excellent examples, such as a colorized explanation for the constant $e,$ which they also provide as a LaTeX document . We can use it to recreate the output on the web:

$\growth \text{The base for continuous growth}$ $\plain \text{is}$ $\unitQuantity \text{the unit quantity}$ $\unitInterest \text{earning unit interest}$ $\unitTime \text{for unit time,}$ $\compounded \text{compounded}$ $\perfectly \text{as fast as possible}.$

To use colors in LaTeX, we must include the color package in the header.

Then, we can define new colors in the body and use them in math mode. For example:

Overleaf has a helpful article about Using colors in LaTeX .

When using Mathjax or KaTeX to render equations on the web, we can use the extended color keywords or RBG hex values for the color argument. The commands must be inside math mode.

We can define colors using the \color command.

Alternatively, we can define the colors using the \textcolor command.

We can take into account color blindness by using a colorblind safe palette. We can use Colorbrewer , a tool for coloring planar graphs (aka maps), to select different colorblind safe palettes. In Colorbrewer, choose the colorblind safe option, select the color scheme from sequential or diverging , the number of data classes and HEX or RGB output. Finally, copy the color codes from the bottom left to the LaTeX commands.

Let us have the following BibTeX entry stored in bibliography.bib file.

We can refer to this entry in the Markdown document using syntax @key_name or [@key_name] . Pandoc creates references at the bottom of the document.

Vector Graphics

Markdown allows inserting vector graphics with the standard syntax.

Using vector graphics when creating PDFs requires Inkscape .

I wrote this article based on my experiences of writing scientific essays and handouts, technical documentation, and blog articles. I hope it helps you to write these types of documents more effectively.

If need resources for faster, more comfortable typing, I recommend the Keyboard section on the Ergonomic Workstation page. For improving your grammar and editing text, you can check out my Digital Writing page.

If you enjoyed or found benefit from this article, it would help me share it with other people who might be interested. If you have feedback, questions, or ideas related to the article, you can contact me via email . For more content, you can follow me on YouTube or join my newsletter . Creating content takes time and effort, so consider supporting me with a one-time donation .

Jaan Tollander de Balsch

Jaan Tollander de Balsch is a computational scientist with a background in computer science and applied mathematics.

How to Teach Scientific Notation

January 13, 2022 rethinkmathteacher.com Math Teaching Resources , Pre-Algebra 0

in 5 Easy Steps

Scientific notation—a relatively straightforward process that so many students struggle with. Why? There may be many reasons, but I think it’s partly because many math teachers are strapped for valuable class time, and they tend to rush over it. 

But there’s a way you can help your students master converting scientific notation into standard numbers, converting standard numbers into scientific notation, and performing operations with numbers expressed in scientific notation – without spending days and days covering it. We’re going to discuss how to these 8th grade math skills in the simplest, fastest way, all while keeping students engaged. 

1. Lay the Foundation

Before you jump in, make sure your students have all the tools they need to learn scientific notation. Spend some time reviewing Place Value, Exponents, and terms like “to the power of.”

Without these foundations, students will needlessly struggle with scientific notation. Check out these worksheets that can help you review:

• Add, subtract, multiply, and divide numbers with decimals
• Exponent Rules: Multiply, Divide, and Power of a Power
• Powers and Exponents
• Negative and Zero Exponents

2. Explain the “What”

Part of the reason many students struggle with scientific notation is that they don’t understand its purpose or value. When students don’t see the value in what they’re learning, they tune it out, and their grades reflect it. 

What is the what?

Basically, scientific notation is a type of shorthand . We use different forms of shorthand every day—abbreviations, symbols, and even emojis can represent words in a shorter form. 

Consider asking your students what texting abbreviations they frequently use. (LOL, SMH, FR, WYD, WDYM come to mind.) Just like we use abbreviations through text, we also use scientific notation to make big or small numbers easy to write and understand.

In short, scientific notation is a method of presenting very large numbers or very small numbers in a simpler form.  

Give the example of pi—a number your students should be familiar with. You can’t write out all the numbers of pi on a sheet of paper, or even every sheet of paper on planet earth. So what do we do? We create a shorthand. 

Instead of writing all the numbers of pi, an impossible task anyway, we use the Greek π symbol. But not every number in the universe can have its own fancy symbol, so we use scientific notation for the rest.

3. Explain the “Why”

First and foremost, mathematicians are lazy. To their credit, scientists are also lazy. Students are lazy, and teachers are lazy, too. In fact, all humans are lazy because we will never do more work than we need to. That’s why, when presented with numbers that have a lot of digits, we want to shorten them as much as possible, to make it easier to read and write. 

Show your students a really long number on the board, such as the mass of the earth. Scientists estimate the mass of the earth to be 6,000,000,000,000,000,000,000,000kg.

Have your students imagine this scenario: their cruel math teacher has decided that students will write essays on their math lessons. Today they will write a 1000-word essay on the mass of the earth. Throughout this essay, they will inevitably have to write the number over and over and over again. They will probably get hand cramps writing all those zeros. 

Using scientific notation, however, they can convert this number into a much simpler form to represent the earth’s mass: 6 × 10 23 .

Ask students to theorize other extremely large numbers, or small numbers, that would be difficult to write – so writing them in scientific notation would be much easier.

Examples of numbers that need to be written in scientific notation:

• the weight of a planet
• the weight of an atom
• the distance from our planet to another galaxy
• the length of a microscopic deep sea creature
• how far Superman could travel if he ran as fast as he could for an hour

4. Explain the “How”

Decimal notation to Scientific Notation

The “regular” version of these long numbers is called decimal notation , as opposed to the shortened scientific notation .

Show your students that scientific notation will always follow a specific format: __ × 10 n . 

When converting to scientific notation, the goal, essentially, is to determine the value of the exponent . To achieve this goal, all we really have to do is count. And tell your students it’s basically that simple. 

Let’s work with another big number. The surface area of all the water on earth is estimated at 140,000,000 square miles. To convert this number to scientific notation, have your students follow these simple steps (you may want them to write these steps down, or have them repeat the steps after you):

1. Locate the decimal.

For some numbers, the decimal may not be written but is understood to be at the end. So for the surface area, we can assume the decimal falls at the end as well (140,000,000.0). For very small numbers (less than 1), the decimal will already be written. 

2. Move the decimal. 

The reason we want to move the decimal is to get rid of all those pesky zeros. But we aren’t moving the decimal just any ol’ place—we have to move it to the perfect location. Always move the decimal after the first digit that is not a zero . In the surface area example, that would be after the 1 (1.400000000). Remind your students that for very large numbers, the decimal moves left, and the exponent will be positive. 

Please Note: It might also be beneficial to explain to your students that when they move the decimal, and get rid of all the zero’s, the new number will be a number greater than one but less than 10.

3. Count the moves. 

As they move the decimal, your students should keep track of how many decimal places they moved it. In essence, our notation is starting at 10 0 . The number of decimal places it moves will become our new exponent.

When you count the “moves” in 140,000,000, have your class count along, saying “ten to the first, ten to the second, ten to the third, ten to the fourth” and so on. Counting aloud helps students recognize how moving the decimal affects the exponent. 

If your students struggle with Place Value, they might struggle here. Consider asking your students how many decimal points it would have to move to fall just behind the 1. This may help you identify if you need further review before moving on. 

You can explain that moving the decimal is like taking it on a road trip, and you need to keep track of all its stops along the way. 

4. Write the notation

The rest is simple. Once they have moved the decimal, they will write the new number without its zeroes. Always follow this simple pattern:

__ × 10 x   

Since they already have their new number, and since they should have already counted how many place values the decimal moved, they have all the components they need to fill in the rest:

1.4 × 10 8   

When working through the first couple of examples, say the steps aloud each time, having your students repeat after you. When working on later examples, ask students for the next steps. If they memorize the steps, they will be more likely to replicate it on their own.

Since they now have the foundations for large numbers, go through an example with a small number. When working on very small numbers, explain to students that they will move the decimal to the right . As a result, the exponent with be negative.

Make sure to emphasize that when you move the decimal of a large number, it will always move left , and the exponent will be positive. When you move the decimal of a small number, it will always move right and the exponent will be negative. (Left, positive; right, negative)

Scientific Notation to Decimal Notation

Once students understand the first process—converting standard notation into scientific notation—the reverse is much easier. And we’re going to follow a similar process, but the goal here is a little different. When converting back to standard notation, our goal now is to bring the value of the exponent down to zero, rather than looking for the value of the exponent. 

Let’s try the following: 7.2 × 10 6

• Move the decimal.

Before, we had a designated place to put the decimal—after the first digit that is not a zero. Now, however, we don’t have a designated location for the decimal. We have to use information in the notation to find the new perfect location. That’s what the exponent is for. 

The value of the exponent equals how many place values your decimal will move. If the exponent is positive , it will now move right. If the exponent is negative , it will now move left . Remind your students this is the opposite of the first process. 

Again, the goal is to get the 10 down to a power of 0. So we’re going to have to move the decimal 6 place values to get down to 0. 

• Bring back the zeros.

As you begin to move the decimal, you will quickly run out of digits. To solve this problem, you will add a 0 for each place value you pass. In such a case, 7.2 × 10 6 will become 7,200,000.

5. Practice, Practice, Practice

The only way to master this process is to practice. Practice a few examples together as a class to build the foundation, having them repeat the steps as you go along. Then begin to ask students for the steps on their own, either as a class or cold call individually. 

Once they display understanding, have them work on a question alone. You can have them work on a question individually at their desks, bring several volunteers to work on a question at the board, or ask a volunteer to go through the whole process in front of the class while the rest follow along. 

However, my favorite strategy is to use learning stations to have students practice the skill we are working on. This allows me to provide immediate feedback to all of my students, and allows me to differentiate. My struggling students are given more time with immediate feedback, while my stronger students, who demonstrated mastery quicker than the rest of the class, are allowed to move on to more complicated skills (like solving operation problems on numbers in scientific notation ).

Save time with the best resources.

The best way for students to master scientific notation is to practice the skill and get immediate feedback. But creating worksheets (and their answer keys!) can take a lot of time—something most math teachers don’t have. 

Luckily for you, I have a plethora of resources of vetted, student-approved worksheets, including my own Scientific Notation Bundle . The bundle includes fact-based practice problems, quizzes with answer sheets, mazes, matching activities, student progress sheets, video tutorial links, and a whole lot more.

You can use this bundle to create learning stations and have students practice these principles in class, or you can easily distribute them to students working remotely. However you use them, be sure to leave a reply letting me know how they worked for you and your class. 

Until then, happy teaching!

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Scientific Notation Calculator

Enter a regular number below which you want to convert to scientific notation.

The scientific notation calculator converts the given regular number to scientific notation.

A regular number is converted to scientific notation by moving the decimal point such that there will be only one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent of 10.

If you move the decimal point to the left, the exponent of 10 will be positive. If you move decimal point to the right, the exponent of 10 will be negative.

Click the blue arrow to submit. Choose "Convert to Scientific Notation" from the topic selector and click to see the result in our Pre-Algebra Calculator!

Convert to Scientific Notation

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