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Standard Form Equation of a line

Examples of equation and Graphs

Overview of different forms of a line's equation

There are many different ways that you can express the equation of a line . There is the slope intercept form , point slope form and also this page's topic. Each one expresses the equation of a line, and each one has its own pros and cons. For instance, point slope form makes it easy to find the line's equation when you only know the slope and a single point on the line. Standard form also has some distinct uses, but more on that later.

Definition of Standard form Equation

$ \text{Formula } : \\ Ax + By = C \\ A \ne 0 \\ B \ne 0 $

General Formula for x and y-intercepts

For the equation of a line in the standard form, $$ Ax + By = C $$ where $$ A \ne 0 $$ and $$ B \ne 0$$ , you can use the formulas below to find the x and y-intercepts.

$ \text{X Intercept: } \\ \frac C A = \frac 6 3 = 2 $

$ \text{Y Intercept: } \\ \frac C B = \frac 6 2 = 3 $

Example and Non Example Equations

When is standard form useful.

• When you want to graph the line.
• When you know the y-intercept of the line .
• Contrast this with slope intercept form -- in this case, you have to do more work to find the x-intercept .

When are other forms more useful?

• Slope intercept form and point slope make it easier to find the slope of your line . In standard form, you must do some work to get the slope .
• Point slope makes it easy to graph the line when you only know the line's slope and a single point or when you know 2 points on the line.

Video Tutorial on Standard Form Equation of a Line

Sample Practice Problems

Find the intercepts and graph the following equation: 3x + 2y = 6

How to find the x-intercept :

How to find the y-intercept :

How to Graph from Standard Form

Plot the x and y-intercepts and draw the line on the graph paper!

Practice Problems

Identify which equations below on the right are in standard form.

• Equation 1: 2x + 5 = 2y
• Equation 2 : 2x + 3y = 4
• Equation 3: y = 2x + 3
• Equation 4 : 4x -$$ \frac 1 2 $$ y = 11

Equation 2 and equation 4 are the only ones in standard form.

Equation 3 is in Slope intercept form .

• Equation 1 : 11 = ¼x + ½y
• Equation 2: 2x + 5 + 2y = 3
• Equation 3: y - 2 = 3(x − 4)
• Equation 4 : $$ \frac 1 2 $$ y − 4x = 0

Equation 1 and equation 4 are the only ones in standard form.

Equation 3 is in point slope form .

Find the intercepts and then graph the following equation 2x + 3y = 18 .

First, find the intercepts by setting y and then x equal to zero. This is pretty straightforward since the line is already in standard form.

Set y = 0 :

Solve for x:

Solve for y:

Plot the x and y-intercepts , which in this case is (9, 0) and (0, 6) and draw the line on the graph paper!

Find the intercepts and then graph the following equation 3x + 5y = 15 .

Plot the x and y-intercepts , which in this case is (5, 0) and (0, 3) and draw the line on the graph paper!

Find the intercepts and then graph the following equation 3y − 2x = -12 .

Plot the x and y-intercepts , which in this case is (6, 0) and (0,-4) and then graph the equation!

• Linear Equations
• Equation of Line Formula
• Slope Intercept Form
• Slope Intercept to Standard Form
• Point Slope to Standard Form
• Worksheet on standard form equation (pdf with answer key on this page's topic)

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Standard Form

Standard form in maths is the method of representing a particular element in the most common manner. From large numbers to small numbers to equations to lines, every element in maths is denoted in a standard form. Let us explore this interesting concept of standard form in various elements of maths such as fractions, equations, algebra, slope along with learning the standard from formula. Solving examples and understanding the basic thumb rule will help in understanding the concept better.

What is Standard Form?

A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules. Representing very large or very small numbers concisely, the standard form is used. For example, 4.5 billion years is written as 4,500,000,000 years. As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few 0’s less or more while writing in the number form. In this case, writing the number in standard form is very helpful. For example, the standard form of 4,500,000,000 = 4.5 × 10 9 . Not only numbers but the fractions, equations, expressions, polynomials, etc also can be written in the standard form.

Let us study the standard form of each of these in detail.

Standard Form of Number

The standard form of numbers (which is also known as " scientific notation " of numbers)  has different meanings depending on which country you are in. In the United Kingdom and countries using UK conventions, the standard form is another name for scientific notation. Scientific notation is the process of writing a very large or very small number using numbers between 1 to 10 multiplied by the power of 10. For example, 3890 is written as 3.89 × 10 3 . These are numbers that are greater than 1 use positive powers of 10. Numbers less than 1 use the negative power of 10. For example, 0.0451 is written as 4.51 × 10 -2 .

In the United States and countries using US conventions, the standard form is the usual way of writing numbers in decimal notation . Using the same example,

Standard form = 3890, Expanded form = 3000 + 800 + 90, and Written form = Three thousand eight hundred and ninety.

Standard Form of Fraction

In the case of fractions, we need to ensure that in the standard form of fractions, the numerator and denominator must be co-prime numbers . That means they have no common factor other than 1, hence the standard form is also called the simplest form of a fraction . For example, 14/22 and 13/6. The simplest form of 14/22 = 7/22 and 13/6 is already in its simplest form as 13 and 6 are co-prime.

Standard Form of Equations

The standard form of an equation is where zero goes on the right and everything else goes on the left. i.e., it is of the form

• Expression = 0.

This helps in solving the equation in a simple manner. Equations used for polynomials, linear and quadratic have a standard form, let us look at what they are.

Standard Form of Polynomial

The standard form polynomial is written with the exponents in decreasing order to make calculations easier. A polynomial is said to be in its standard form, if it is expressed in such a way that the term with the highest degree is placed first, followed by the term which has the next highest degree, and so on. There are two very simple rules of writing a polynomial in a standard form, they are:

• Write the terms in the descending order of their powers (also called exponents).
• Ensure the polynomial contains no like terms.

Hence, the standard form is a n x n  + a n-1 x n-1  + a n-2 x n-2  + ... + a 1 x 1  + a 0 . For example, the standard form of equation y 2 + 7y 6 - 8y - y 9 is written as -y 9 + 7y 6 + y 2 - 8y.

Note: The thumb rule for writing a polynomial in its standard form is D-U. D stands for Descending and U stands for unlike terms.

Standard Form of a Linear Equation

The standard form of linear equations also known as the general form is a method of writing linear equations. A linear equation can be written in different forms like the standard form, the slope-intercept form, and the point-slope form . The standard form of a linear equation is expressed in two ways, with one variable and with two variables . The standard form of linear equation with one variable is expressed as ax + b = 0 where a and b are constants and the letters x is the variable. The standard form of a linear equation with two variables is expressed as ax + by = c. where a, b, and c are real numbers and the letters x and y are the variables. Look at the image below.

Let’s see how to convert the two lines into the standard form of a linear equation ax + by = c.

Line 1: x + y = 7 i.e. 1x +1y = 7. Here, a = 1, b =1 , c = 7

Line 2: y = 3x i.e. 3x -1y = 0. Here, a = 3, b = -1 , c = 0

Therefore, what we see here is the standard form of equation which is linear i.e. ax + by = c.

Standard Form of Slope of a Line

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. Representing a line geometrically, we use the standard form of a linear equation (mentioned above). To determine the slope of a line that is expressed graphically, the equation must be converted to a slope-intercept form . To do this, we must solve the equation for y, and the standard form of a slope is expressed as y = mx + c, where m is the slope of the line. This formula is used when the line is straight.

When there are two points in a plane , the slope can be defined as the ratio of change in the value of y to the change in the value of x. The standard form of slope of a line is expressed as m = (y 2 – y 1 )/(x 2 – x 1 ). The image below represents both the coordinates on a graph.

Standard Form of Quadratic Equations

The standard form of quadratic equation in a variable x is of the form ax 2 + bx + c = 0, where a ≠ 0, and a, b, and c are real numbers . Here, b and c can be either zeros or non-zero numbers and

• 'a' is the coefficient of x 2
• 'b' is the coefficient of x
• 'c' is the constant

Apart from the standard form of a quadratic equation, a quadratic equation can be written in several other forms.

• Vertex Form: a (x - h) 2 + k = 0
• Intercept Form: a (x - p)(x - q) = 0

A parabola is a graph of a quadratic function that refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola . The following graphs are two typical parabolas; their x and y-intercepts are marked by green dots, and the vertex of each parabola is marked by a blue dot.

The graph will have one of the two shapes as shown above.

• When a > 0, it represents the first parabola (figure 1) which opens upward (U-shaped).
• When a < 0, we obtain a parabola that opens downward (inverted U-shaped) as shown in figure 2.

When we put y = 0 in the above equation, we get the x-intercept which is also called the roots of the equation . Finding the roots provides us with the turning point of the parabola and the vertical line which is drawn from the turning point is called the axis of symmetry . Look at the image below for reference.

Standard Form Formula

The standard form formula represents the standard form of an equation which is the commonly accepted form of an equation. The formula to represent the standard form formula is based on the degree of the equations.

The standard form of a linear equation is the basic form of an equation. The standard form of a linear equation with two variables and more than two variables is presented below. Here x, y, or x 1 , x 2 , x 3 ,.... represent the variables and a, b, a 1 , a 2 , a 3 , ........ a n are referred to as the coefficients. The numerics placed to the right of the equals to sign are called constants.

ax + by = c

a 1 x 1 +a 2 x 2 + a 3 x 3 + ........ + a n x n = D

The standard form of a quadratic equation is a second-degree equation and has a variable, coefficients, and constant term. Here it is a single variable x of degree 2. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a ≠ 0.

Further, we have standard form formulas for equations of higher degrees. Also in coordinate geometry , we have a standard form for different geometric representations such as a straight line, circle , ellipse , hyperbola , and parabola.

• Straight-line: ax + by = c, where a is a positive integer , and b, and c are integers.
• Circle: (x - h) 2 + (y - k) 2 = (r) 2 , where ( h, k) is the center and r is the radius .
• Ellipse: x 2 /a 2  + y 2 /b 2  = 1
• Hyperbola: (x-x 0 ) 2 /a 2  - (y-y 0 ) 2 /b 2  = 1, where x 0 , x 0 are the center points, a = semi-major axis and b = semi-minor axis.
• Parabola: (x - h) 2 = 4p(y - k)

☛Related Topics

Listed below are a few topics that are related to a standard form.

• Standard Form Calculator
• Standard Form to Vertex Form
• Polynomial in One Variable in Standard Form

Standard Form Examples

Example 1: Liza is trying to find out which of the following equations represent the given graph. As there are no values of the coordinates given, she is not able to decide. How can we use the standard form concept to solve her problem?

1. x 4 + 7x 2 - 5x = 4

2. 4x + 5y = 0

3. y 2 + 7y 6 - 9y = y 3

4. 3z 4 + 7z 5 = -12z

The graph given here represents a line . Now, a line represents a linear equation in two variables which has a degree of 1. Out of the four equations given above, only option (b) is linear.

Answer:  Therefore, the graph represents the equation, 4x + 5y = 0.

Example 2: Anna showed her class notes on polynomials to her teacher. The teacher returned her notes with a remark "Write the polynomial x 2 -10x + 16- x 2 + x 5 - 3x 4 + 3x 2 in standard form." What is the correct format Anna should have used?

To write a polynomial in standard form, two rules must be taken care of.

1. Write the terms in descending order of their powers.

2. All terms must be unlike.

Let us first arrange in descending order .

x 2 - 10x + 16- x 2 + x 5 - 3x 4 + 3x 2 = x 5 - 3x 4 + x 2 - x 2 + 3x 2 - 10x +16

After adding like terms , we get

x 5 - 3x 4 +3x 2 - 10x + 16.

Answer:  Therefore, the standard form is x 5 - 3x 4 +3x 2 - 10x + 16.

Example 3: Convert the following quadratic equation into standard form: (1/x) + x = 1.

(1/x) + x = 1

Multiplying with x on both sides,

1 + x 2 = x

Shifting R.H.S. terms to L.H.S.,

x 2 - x + 1 = 0

Answer:  Therefore, Standard form of the given quadratic equation is x 2 - x + 1 = 0.

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Practice Questions on Standard Form

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FAQs on Standard Form

What is the definition of standard form in math.

Standard form in math is the method of representing a particular element (numbers, fractions, equations, etc) in the most common way. Very large numbers or very small numbers are expressed in the standard form. Mathematical elements such as equations are expressed in a standard form to better solve the problem. In other words, a standard form is a form of writing a given mathematical concept like an equation, number, or expression in a form that follows certain rules.

How Do We Write Standard Form in Maths?

The process of writing a given mathematical concept like an equation, number, or expression in certain rules is called the standard form. Depending upon which mathematical concept we are dealing with, the standard form formula will vary.

How Do You Write Standard Form of Number?

Standard form in math of numbers is written differently depending on the country. In the UK, numbers that are greater than 1 use positive powers of 10, and numbers less than 1 use the negative power of 10. For example, 3670000 is written as 3.67 × 10 6 and 0.0763 is written as 7.63 × 10 -2 . This is commonly known as scientific notation of numbers.

What is Standard Form Formula?

The standard form formula refers to the formula presenting the general representation for different types of notation. For example, the standard form of

• a linear equation is ax + by = c.
• a quadratic equation is ax 2  + bx + c = 0
• a cubic equation is ax 3  + bx 2  + cx + d = 0

What is the Standard Form Formula for Parabola?

The standard form formula of the equation of the parabola is this: (y - k) 2 = 4p(x - h), where p≠ 0 only in case a parabola has a horizontal axis.

• The vertex of this parabola is at (h, k).
• The focus is at (h + p, k).

What is the Standard Form for Slope Formula?

The standard form of the  slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line whereas the standard form of a linear equation is Ax + By = C and the slope in this form is -A/B.

How to Use Standard Form Formula?

We can use the standard form formula depending on the equation, if it's linear, quadratic, etc. Just rewrite the given formulas in the standard form.

• Standard form of linear equation : ax + by = c
• Standard form of a quadratic equation is a second degree equation: ax 2 + bx + c = 0

What is the Standard Form of Equation?

In the standard form of an equation, 0 is usually present on the right side, whereas the rest of the expressions are on the left. Also, the terms are arranged in the descending order of their exponents. To convert an equation into stadard form, just apply arithmetic operations on both sides to turn the right side to be 0.

Standard Form

What is "Standard Form"?

that depends on what you are dealing with!

I have gathered some common "Standard Form"s here for you..

Note: Standard Form is not the "correct form", just a handy agreed-upon style. You may find some other form to be more useful.

Standard Form of a Decimal Number

In Britain this is another name for Scientific Notation , where you write down a number this way:

In other countries it means "not in expanded form" (see Composing and Decomposing Numbers ):

Standard Form of an Equation

The "Standard Form" of an equation is:

(some expression) = 0

In other words, "= 0" is on the right, and everything else is on the left.

Example: Put x 2 = 7 into Standard Form

x 2 − 7 = 0

Standard Form of a Polynomial

The "Standard Form" for writing down a polynomial is to put the terms with the highest degree first (like the "2" in x 2 if there is one variable).

Example: Put this in Standard Form:

3 x 2 − 7 + 4 x 3 + x 6.

The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

x 6 + 4 x 3 + 3 x 2 − 7

Also, within each term, it is nice to have the variables in alphabetical order (if it does not make things more confusing):

yzx 2 + 4 yx 3

The highest degree is 3, so that goes first, also put the variables in alphabetical order

4 x 3 y + 3 x 2 yz

Standard Form of a Linear Equation

The "Standard Form" for writing down a Linear Equation is

Ax + By = C

A shouldn't be negative, A and B shouldn't both be zero, and A , B and C should be integers.

Bring 3x to the left:

−3x + y = 2

Multiply all by −1:

3x − y = −2

Note: A = 3, B = −1, C = −2

Ax + By + C = 0

is sometimes called "Standard Form", but is more properly called the "General Form".

Standard Form of a Quadratic Equation

The "Standard Form" for writing down a Quadratic Equation is

( a not equal to zero)

Expand "x(x−1)":

x 2 − x = 3

Bring 3 to left:

x 2 − x − 3 = 0

Note: a = 1, b = −1, c = −3

Standard Form of a Circle Equation

With a circle like this:

The Standard Form is this:

(x−a) 2 + (y−b) 2 = r 2

See Circle Equations for more details.

Standard Form: Worksheets with Answers

Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. And best of all they all (well, most!) come with answers.

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Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers.

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Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. It really is one of the very best websites around.

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Standard Form

Here we will learn about standard form including how to convert between ordinary numbers and standard form, and how to calculate with numbers in standard form.

There are also standard form worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is standard form?

Standard form is a way of writing very large or very small numbers by comparing the powers of ten. It is also known as scientific notation.

Numbers in standard form are written in this format:

Where a is a number 1 ≤ a < 10 and n is an integer.

To do this we need to understand the place value of a number

Let’s look at the number 8290000 and write the digits in a place value table,

So 8290000 written in standard form is

Standard form notation

Standard form notation is a representation of place value which compliments the decimal number system, as shown in the table below.

Any integer or terminating decimal can be written using the standard form notation a\times10^{n} . The table below shows how the value of a can remain the same while the power of ten ‘n’ changes the place value of those digits.

Using standard form notation enables us to write very large or very small numbers.

Using standard form notation also enables us to compare the size of very large or very small numbers easily. 

Which is larger, 8560000000000 or 45320000000000?

At a glance it is difficult to tell which is larger, but written in standard form you can compare these numbers very quickly as shown. 

Which is larger, 8.56\times10^{12} or 4.532\times10^{13} ?

Instantly we can see that 4.532\times10^{13} is the larger number as it has the higher power of ten. 

Standard form calculation

When numbers are written in standard form it can make some calculations neater and quicker to compute. This is particularly the case for calculations involving multiplication, division and powers. However, this is reliant on an understanding of the rules of indices. 

Multiplication rule a^{b} \times a^{c} = a^{b+c}

Division rule a^{b} \div a^{c} = a^{b-c}

Zero power rule a^{0} = 1

Power of power rule (a^{b})^{c} = a^{b \times c}

To review this topic see step-by-step guide : Laws of indices

340000 \times 0.002

In standard form this changes to (3.4 \times 10^{5}) \times  (2 \times 10^{-3})

Brackets are often used to show each separate number in standard form but they are not actually necessary 3.4 \times 10^{5} \times 2 \times 10^{-3}

As multiplication is commutative (the order does not matter) we can reorder this calculation to 3.4 \times 2 \times 10^{5} \times 10^{-3}

3.4 \times 2 = 6.8

10^{5} \times 10^{-3} = 10^{5+-3} = 10^{2} (applying the multiplication rule of indices a^{b} \times a^{c} = a^{b+c})

(3.4 \times 10^{5}) \times  (2 \times 10^{-3}) = 6.8 \times 10^{2}

340000 \times 0.002 = 680

Let’s look at how to calculate with standard form in more detail.

How to calculate with standard form

In order to write numbers in standard form.

• Identify the non-zero digits and write these as a decimal number which is greater than or equal to 1 but less than 10
• In order to maintain the place value of the number, this decimal number needs to be multiplied by a power of ten
• Write the power of ten as an exponent

Write your number in standard form

E.g. Convert 4500 to standard form.

• Writing 4500 as a decimal between 1 and 10 is 4.5
• 10 3 = 1000
• So 4500 written in standard form is 4.5 × 10 3

In order to convert from standard form to ordinary numbers

• Convert the power of ten to an ordinary number
• Multiply the decimal number by this power of ten
• Write your number as an ordinary number

E.g. Convert 7.1 × 10 4 to standard form.

• 10 4 = 10000
• 7.1 × 10000
• So 7.1 × 10 4 as an ordinary number is  71000

Step-by step guide: Converting to and from standard form

In order to multiply and divide in standard form

• Multiply or divide the non-zero numbers
• Multiply or divide the powers of ten by adding or subtracting the indices
• Write the solution in standard form, checking that the first part of the number is 1 ≤ x < 10

E.g. Work out (4 × 10 5 ) × (2 × 10 8 )

• 10 5 × 10 8 = 10 13
• 8 × 10 13 is in standard form

E.g. Work out (2 × 10 3 ) ÷ (4 × 10 8 )

• 2 ÷ 4 = 0.5
• 10 3 ÷ 10 8 = 10 -5
• 0.5 × 10 -5 is not in standard form

0.5 is less than 1 so multiply 0.5 by 10 which gives the result 5 .

To compensate, you need to divide the power of ten by 10 which has the effect of subtracting one from the power, which gives the result 10 -6 .

So the answer is 5 × 10 -6 .

Step-by-step guide: Multiplying and dividing in standard form

In order to add and subtract numbers in standard form

• Convert the number with the lower power of ten so that both numbers have the same power of ten
• Add the non-zero integers
• Check your answer is in standard form

E.g. (3 × 10 5 ) + (6 × 10 4 )

• 6 × 10 4 = 0.6 × 10 5
• 3 + 0.6 = 3.6
• 3.6 × 10 5 is in standard form

E.g. (9 × 10 4 ) − ( 3 × 10 3 )

• 3 × 10 3 = 0.3 × 10 4
• 9 − 0.3 = 8.7
• 8.7 × 10 4 is in standard form

Step-by-step guide: Adding and subtracting in standard form

Explain how to calculate with standard form

Standard form worksheet

Get your free standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Standard form examples

Example 1: writing numbers in standard form.

Write this number in standard form:

• The non-zero digits need to be written as a decimal number

The number needs to lie between 1 ≤ x < 10

So the number will begin as 5.2 …

2 You now need to maintain the value of the number by multiplying that decimal by a power of ten 

3 Write that power of ten as an exponent

4 W rite your number in standard form

Example 2: writing a number in standard form as an ordinary number

Write 9.4\times10^5 as an ordinary number.

Write the exponent as a power of ten

Multiply the decimal number by that power of ten

Write your answer as an ordinary number

Example 3: writing a small number in standard form

Write 0.0068 in standard form.

The non-zero digits need to be written as a decimal number. The number needs to lie between 1 ≤ x < 10

So the number will begin as 6.8

Identify what power of ten the decimal needs to be multiplied by in order to preserve place value

Write that power of ten as an exponent

Example 4: multiplying numbers in standard form

Calculate (5\times10^{4}) \times (7\times10^{8}) .

Write your answer in standard form.

Multiply the non-zero digits

Multiply the powers of ten by adding the powers

Put these two parts together

However, this number is not in standard form as 35 is not a decimal number.

To convert 35 to a decimal between 1 and 10 you need to divide it by 10 . To maintain the value of the number, you need to multiply the power of ten by 10 , which adds one to the exponent.

Example 5: dividing numbers in standard form

Calculate (8\times10^{7})\div(2\times10^{2}) .

Divide the non-zero digits

Divide  the powers of ten by subtracting the powers

Check your number is in standard form

This is in standard form as 4 is greater than 1 but less than 10 .

Example 6: adding numbers in standard form

Calculate (6\times10^{4})+(2\times10^{3}) .

 Convert the number with the smaller power so that both numbers have the same power

You need to increase the power, to do this multiply it by 10 to add one to the power.  But to maintain the value of the number you need to divide the non-zero number by 10 .

 You will now have two numbers with the same exponent

6\times10^4 and 0.2\times10^4

You now add the two non-zero numbers

The exponents are the same after what you did in step 1

So your answer in standard form is 6.2\times10^4 .

Example 7: subtracting numbers in standard form

Calculate (5\times10^{4})−(4\times10^{3}) .

Convert the number with the smaller power so that both numbers have the same power

To do this identify the number with the lowest exponent.

You need to increase the power. To do this multiply it by 10 to add one to the power. But to maintain the value of the number you need to divide the non-zero number by 10 .

5\times10^{4} and 0.4\times10^{4}

So your answer in standard form is 4.6\times10^4 .

Common misconceptions

• Writing a number with the incorrect power for a large or small number

This can happen by counting the zeros following the first non zero digit for large numbers or zeros after the decimal point for small numbers, then writing this as the power.

• Identifying incorrect place value with small numbers

For example, in a number such as 0.000682 , selecting the ‘2’ to determine the power rather than the ‘6’ which has a higher place value can lead to …

• Not converting solutions to standard form

After calculating with standard form, a common mistake is to not check that the first part of the number is 1 ≤ a < 10 .

• Not giving the solution in the correct form

It is important to check what form the question asks for the solution in – ordinary number or standard form.

Practice standard form questions

1. Write 86,000 in standard form.

The number between 1 and 10 here is 8.6. Since 8 is in the ten thousands column,

2. Write 0.0097 in standard form.

The number between 1 and 10 here is 9.7. Since 9 is in the thousandths position,

3. Write 5.9 \times 10^{3} as an ordinary number.

4. Work out (6\times10^{8})\times(3\times10^{4}) . Write your answer in standard form.

However  18 \times 10^{12} is not in standard form since 18 is not between 1 and 10.

So we need to divide 18 by 10 and, to compensate, increase the power of ten by 1.

This gives us

5. Work out (9\times10^{7}) \div (4\times10^{2}) . Write your answer in standard form.

This is already in standard form since

6. Work out (7\times10^{5})+(2\times10^{4}) . Write your answer in standard form.

(increase the power of ten by 1 and divide the 2 by 10 to compensate)

Standard form GCSE questions

1.   The distance from Earth to Mars is 2.88\times10^8km , to three significant figures. Write this number in standard form.

288, 000, 000 km

2. (a) The speed of light is 3\times10^8m/s , rounded to one significant figure. Write this as an ordinary number.

(b) The mass of Earth is 5.97\times10^{24}kg . Work out the mass of the Earth in grams. Write your answer in standard form.

300, 000, 000 m/s

Recognising 1000g in a kilogram or multiplying by 1\times10^3

5.97\times10^{27} g

3.   Put these numbers in order. Start with the smallest number.

62\times10^{-2}, \quad \quad 0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607

Converting all of the numbers to the same form for comparison or orders 3 of the four numbers correctly.

0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607, \quad \quad 62\times10^{-2}

Learning checklist

You have now learned how to:

• Write numbers in standard form
• Calculate with numbers in standard form

The next lessons are

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• Quadratic equations

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Working with standard form in chemistry

By Emily Rose Seeber 2020-02-25T11:12:00+00:00

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Help your students apply their maths knowledge to solving science questions

Source: © Getty Images

Mastering standard form and its use in chemistry can be a real balancing act

Students meet standard form in maths, but this form of numerical notation is critical for calculations in chemistry for the 14–16 and post-16 age groups. Generally, chemistry teachers presuppose that students have mastered decoding standard form notation by the time they need to use it in chemistry contexts, but this is not always the case. In maths, standard form is typically localised in questions in which students’ ability to manipulate quantities in standard form is the assessment criteria. In chemistry, using standard form accurately is a prerequisite for solving problems and may not independently carry any method marks. This means that standard form adds a level of cognitive challenge to a problem rather than being a problem in itself. As a result, a student able to use standard form correctly in maths will not always get it right in chemistry.

Maths lessons tend to focus on the ‘moving the decimal point’ (or ‘moving the digits’) method of converting numbers into and out of standard form.

For example

To convert 0.00357 into standard form, the decimal point is moved three spaces to the right (or three digits jump left over the decimal point) to give 3.57 (so the only integer before the decimal point is a number between 1 and 9), and the 10 -3 shows how far and in which direction the decimal point moved (or the numbers jumped), giving an answer of 3.57 x 10 -3 .

Maths lessons tend to focus on the ‘moving the decimal point’ (or ‘moving the digits’) method of converting numbers into and out of standard form. For example,

To convert 0.00357 into standard form, the decimal point is moved 3 spaces to the right (or 3 digits jump left over the decimal point) to give 3.57 (so the only integer before the decimal point is a number between 1 and 9), and the 10 -3  shows how far and in which direction the decimal point moved (or the numbers jumped), giving an answer of 3.57 x 10 -3 .

While this method is incredibly useful for computing these conversions, it hides the purpose of using standard form in the first place: expressing magnitude. Standard form allows scientists to compare the order of magnitude of different quantities in a way that helps them to explain phenomena and make useful predictions.

Essentially, there are two challenges we face from students when using standard form in chemistry classrooms: ‘can’t’ and ‘won’t’. The ‘can’t’ students struggle to interpret significant figures and so make mistakes in their use: they have not yet attained mastery of the principles. The ‘won’t’ students don’t see the point of using standard form as it takes time for them to convert in and out of the notation and/or they can just use a calculator. They have not yet understood the relationship between standard form and orders of magnitude. There is often a significant overlap between the ‘can’ts’ and ‘won’ts’.

Interpreting standard form

Part of why students struggle with standard form is that they cannot ‘read’ numbers in standard form as fluently as numbers written out in full as they are not exposed to them so regularly. But, mastery of converting in and out of standard form is absolutely essential for students to deal with the increased cognitive challenge of using standard form in a chemistry context.

Download this

Diagnostic exercises to assess your students’ fluency with standard form as MS Word or pdf .

Download all

Diagnostic exercises to assess your students’ fluency with standard form from the Education in Chemistry website:  rsc.li/3cgv7Sq

Before introducing your students to chemistry topics involving orders of magnitude, assess their ability to deal with it. Download the diagnostic exercises and use the first to help you determine if your students are struggling with fluently decoding the standard form of notation. Depending on how students respond, you will need to follow one of these courses in your lesson:

A.  If most/all students get the right answer, then seat any who struggled with a student who answered confidently and correctly and continue with the lesson using standard form throughout to reinforce notational fluency. Start the following lesson in a similar way.

B.  If students are split 50/50, put them in mixed attainment pairs and ask the stronger students to explain their reasoning to the weaker student. Then ask each pair to design a new diagnostic question and rotate these around the room for students to have a second attempt. If there’s not sufficient improvement, move on to C.

C.  If most/all students get the wrong answer, you’ll need to do some whole-class work on standard form to teach students how to decode the notation more explicitly. Here are some strategies to get you started.

Teaching strategies

One of the weaknesses of the ‘move the decimal point’ method students meet in their maths lessons is that they never relate the numbers they are using to physical quantities. A simple exercise to carry out with 11–14 and 14–16 age groups is to plot a number line for the orders of 10: 1 mm, 10 mm, 100 mm, 1000 mm, 10,000 mm (across the lab) and 100,000 mm (on the playing field), and then compare these to standard form notation. Students experiencing the changes in magnitude themselves helps challenge the misconception that the orders of 10 form a linear scale and demonstrates to students how much the power of 10 matters to the answer.

Another strategy to familiarise students with decoding is to put a numberline on the board and the orders of 10 in a line along it (discussing scale with students), eg 0.001, 0.01, 0.1, 1, 10, 100, 1000. Model writing 1 and 10 in standard form (1 x 10 0 and 1 x 10 1 ) and show your students how these two give the original numbers (10 0 = 1 and 10 1 = 10). Challenge students to spot the pattern and complete the number line in standard form. Finally, ask them to contribute as many different ways of writing the numbers on the number line as they can to assess their progress. If your students need more practice, use a different number line and repeat.

Lastly, expose students to numbers in standard form as often as possible. This means editing questions to express quantities in standard form on a regular basis, even when standard form is not one of the target skills of the lesson. Giving students worked answers (using standard form) and asking them to spot the errors is an excellent way to build in and assess this skill without overdoing calculations.

Why is standard form so useful?

Another significant challenge for chemistry teachers is convincing our students that standard form actually makes problem solving easier. Essentially, students have a misconception that using standard form takes more effort than writing out numbers in their entirety – demonstrating the efficiency of standard form in problem solving is key to engaging the ‘wont’s’. For many of the problems students face in maths, the original number and the standard form notation can both be written out fairly easily: not so in chemistry (6.02 x 10 23 vs 602000000000000000000000). This means chemistry teachers have an opportunity to reinforce students’ mathematical understanding of magnitude in a way that is difficult for maths teachers.

The second diagnostic exercise is designed to help determine how comfortable post-16 students are with multiplying and dividing base 10 indices in a chemistry context to determine the order of magnitude of a quantity. None of the problems are easy, and so pairing the students up or working in small groups may be more productive, circulating to check all students are contributing.

After the pairs/groups have their answers, try these follow-ups:

• Ask groups to come up and present their solutions to one of the problems, explaining any approximations they made, and their choice of notation.
• Ask groups to pick a problem and calculate their answer using standard form and by writing out all numbers in full. Ask groups to demonstrate this on the board, and ask them how standard form makes the problem easier or harder.
• Ask students to repeat the problems using a calculator and see which group made the most accurate initial approximations.
• Ask students to read ‘A Mole of Moles’ from the book What if? and compare their methodology and the kinds of approximations in the article.
• Ask groups to design their own estimation problem. They will need internet access or data books. Offer them bonus points for designing a problem with an answer in similar orders of magnitude to the problems in the diagnostic question.

Mastery of standard form is partly about repeated exposure to the skill, building fluency and automaticity. But it is also about being able to understand what standard form is and why it matters in chemistry. As chemistry teachers, we have the opportunity to deepen students’ mathematical understanding, as we can expose them to extremely large and small numbers in contexts which are familiar to them, in order to solve concrete, tangible problems.

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Standard Form – Definition with Examples

What is standard form, examples of standard form of numbers, real-life examples of standard form, solved examples of standard form, practice problems of standard form, frequently asked questions of standard form.

Did you know that Earth is 4,543,000,000 years old? We know you skipped past all those zeroes, so we will give you a simpler way of saying it: Earth is 4.543 billion years old. Do you see how reading numbers a certain way makes them easier to understand? That’s why all the mathematicians in the world decided to agree on some rules on writing mathematical concepts so that it is convenient for everyone to read, write, and work with. This particular way is called the standard form.

All of the things you see in math, like numbers or fractions or equations or expressions , have a standard form defined for them. We can think of the standard form as the most common way of representing a mathematical element . 

Let’s look at the standard form of some common mathematical elements:

Whole Numbers

You can define the standard form of a whole number as follows.

Any number that we can write as a decimal number , between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

For example, take the number 123,000,000; an easier way of writing this number is ​​

1.23 × 10 8 ;

If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × 10 8 .

• 14,300,000 in the standard form is 1.43 × 10 7 .
• 3000 in the standard form is 3 × 10³.
• Some other examples are 1.98 ✕ 10¹³ and  0.76 ✕ 10¹³.

Factoid: “Standard Form” is also called “Scientific Notation” depending on the mathematical lingo of the country you are in. In the United Kingdom and the countries that follow the same conventions as the United Kingdom, the term “Scientific Notation” is most commonly used whereas in countries that follow the US conventions majorly refer to this form of writing as the “Standard Form”.

Example 1: Consider the number 81,900,000,000,000.

Step 1: Write the first number: 8 .

Step 2: Add a decimal point after this and write the remaining non-zero numbers: 8.19

Step 3: Now count the number of digits after 8. There are 13 digits. This 13 will be the power of 10 while writing the given number in standard form.

Step 4: So, in standard form: 81,900,000,000,000 is 8.19 × 10¹³.

Related Worksheets

• The distance between the Sun and Mars is 141,700,000 miles or 228,000,000 km.

         This distance can easily be written in standard form as: 1.417 × 108 miles or 2.28 × 108 km.

• Atoms are tiny units of matter composed of three fundamental particles—proton, neutron, and electron. A proton and a neutron weigh equally, 1.67 × 10–27 kg. The weight of an electron is 9.11 × 10–31 kg.

Many other quantities, such as the size of planets, the size of microorganisms, the size of microchips, and the country’s population are all expressed in standard form.

A fraction is said to be in standard form when both the numerator and denominator are co-prime numbers.

Factoid: Two numbers are said to be co-prime when their only common divisor(or factor ) is 1. For example, 2 and 3, 4 and 9, 6 and 13. By virtue, two prime numbers are always co-prime.

For example, the fraction  ⅚  is a standard fraction because the numerator 5 and denominator 6 do not have only 1 as their common divisor , whereas the fraction 4/8 is not a standard fraction because the numerator and denominator have common divisors other than 1, i.e., 2 and 4. But no worries, do you want to turn a fraction into its standard form? We got your back. Let’s look at an example.

Consider the fraction 12/20. Step 1: Find all the common divisors of the numerator and the denominator.             In this case, 12 and 20 have common divisors: 1,2,4.

Step 2: Divide the numerator and denominator by their greatest common divisor. Like this:

No Step 3 is required! As you can see ⅗ is a standard fraction and is the standard form of 12/20.

Examples of Standard Form of Fractions

• The most famous standard fraction is 22/7, also called “pi”.
• All unit fractions , i.e., fractions with numerator 1 are standard.

Decimal Numbers

The definition for the standard form of decimal numbers is the same as that of whole numbers,

Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

The scoop about the standard form of decimal numbers lies in their steps. This is a bit different from whole numbers, where you may have noticed, the power of 10 was a positive number like in 1.23 × 10 8 . Let’s see how we can convert a scary-looking decimal number into a pretty-looking decimal number.

Consider the decimal number 0.0004789.

Step 1: Make your decimal number jump from its original position and place it after the first non-zero digit: 4.789 .

Step 2: Count the number of digits you jumped. In this case, we jumped 4 digits. This number will be the power of 10 while writing our decimal number in standard form.

Step 3: If you have made jumps in the direction to the right, the power of 10 will be negative. If you made the jumps in the direction to the left, the power of 10 will be positive.

Step 4: So, in standard form: 0.0004789 is 4.789 × 10 -4 .

Using the same method, the standard form of the decimal number 981.23 will be 

9.8123 × 10 2 .

The standard form of mathematical elements allows us to work in a convenient and efficient manner by making them easily readable for everyone. There is no shortcut to finding the standard form of anything, but here are a few tips and tricks that we have summarized for you that you must keep in mind while writing something in standard form.

Tips to Master Standard Form

• Thoroughly understand the two parts of the standard form: digits and the power of 10.
• Remember that the power is negative if you have made jumps to the right and the power is positive when you have made jumps to the left while putting a decimal number in its standard form.
• Count the places you have moved the decimal point twice before writing the final answer.
• Don’t forget about the second cousin of prime numbers: the co-primes. They are the basics of putting a fraction in its standard form.

Express 321,000,000 in scientific notation.

The scientific form of the number is 3.21 x 10 8 .

Express 0.00005432 in the standard form.

The scientific form of the number is 5.432 x 10 -5 .

Write 25/40 in the standard form.

The scientific form of the given fraction is ⅝.

Standard Form

Attend this Quiz & Test your knowledge.

What is the standard form of the number 78,980,000?

What is the standard form of the number 32145.222, what is the standard form of the fraction 12/18.

Is the standard form the same as a decimal form?

Yes. The standard form, decimal form, and scientific notation of a number are the same.

How do you convert a fraction into its standard form?

A fraction can be converted into standard form by dividing the numerator and denominator by their greatest common divisor.

What is the basic rule of writing standard form?

While writing decimals in scientific notation, standard form, or decimal form, move the decimal place to the left or right until you reach a number from 1-10.

RELATED POSTS

• Roman Numerals – Definition with Examples
• Number System – Definition, Examples, Facts, Practice Problems
• Fraction Less Than One – Definition With Examples
• Dividing Fractions – Definition with Examples
• Fraction Greater Than 1 – Definition with Examples

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GCSE 9-1 Exam Question Practice (Standard Form)

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

Last updated

17 January 2019

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Forms and Templates

Downloads for A3 problem solving, standard work , project management, and value stream mapping .

Problem Solving Templates

A3 Action Plan Form (from Getting the Right Things Done)

• The action plan template helps define the who, what, when, where, and how of a plan on one page.
• Helps track progress and highlight problems so action can be taken.

A3 Status Review Form (from Getting the Right Things Done)

• Top box provides an overview with respect to our critical end-of-pipe metrics.
• Second box provides an overview of activities, and usually reflects what’s been prescribed on the action plan of the right side of the strategy A3.

A3 Strategy Form (from Getting the Right Things Done)

• A strategy A3 is a one-page storyboard on 11-inch by 17-inch paper that helps tell the strategy “story.”
• Logic flows from top left to bottom right, and each box leads to the next one.

Detailed A3 Template (from Managing to Learn)

• Print this A3 template out to remind you of each section of the problem-solving A3 as you are creating your own.

PDSA A3 Template (from On the Mend)

• A3 Template, in Excel, following the PDSA cycle.

Problem Definition Worksheet

• This worksheet can help you breakdown the problem into a clearly defined gap as well as see how the problem aligns to the needs of the business or your True North purpose.

Root Cause Template

• This template gives you space to record the problem as well as the direct causes and underlying causes.

Four Types of Problems

Art Smalley

Managing to Learn: Using the A3 management process

Perfecting Patient Journeys

Beau Keyte , Tom Shuker and Judy Worth

Getting the Right Things Done

Pascal Dennis

Standard Work Templates

Standard Work Operator Balance Chart (OBC)

• The operator balance chart helps create continuous flow in a multi-step, multi-operator process by distributing operator work elements in relation to takt time.

Standard Work Process Study Sheet

• The Process Study Sheet is used to define and record the time for work elements in a process.

Standard Work Production Analysis Board

• A Production Analysis Board is a display that must be located at the exit of the cell (or the line) to show actual performance compared with planned performance on an hourly basis.

Standard Work Skills Training Matrix

• The Skills Training Matrix shows the required and attained skills of every operator.
• The training schedule also should be shown.

Standardized Work Chart

• The standardized work chart shows operator movement and material location in relation to the machine and overall process layout.
• It should show takt time, work sequence, and standard WIP.

Standardized Work Combination Table

• The standardized work combination table shows the combination of manual work time, walk time, and machine processing time for each operation in a production sequence.
• This form is a more precise process design tool than the Operator Balance Chart.
• It can be very helpful to identify the waste of waiting and overburden, and to confirm standard.

Standardized Work Job Instruction Sheet

• The job instruction sheet is used to train new operations.
• It lists the steps of the job, detailing any special knack that may be required to perform the job safely with utmost quality and efficiency.
• It can also be useful for experienced operators to reconfirm the right operations.

Standardized Work Process Capacity Sheet

• The Process Capacity Chart is used to calculate the capacity of each machine to confirm true capacity and to identify and eliminate bottlenecks.
• Processing capacity per shift will be calculated from the available production time, completion time, and tool-change time (and other factors as necessary) for each work piece.

Waste Walk Template (from Perfecting Patient Journeys)

• Taking a “waste walk” is one way to make the waste visible again.
• A waste walk is simply a planned visit to where work is being performed to observe what’s happening and to note the waste. It differs from go-see activities in that you are specifically looking for waste.

Kaizen Express

Toshiko Narusawa and John Shook

Lean Lexicon 5th Edition

Lean Enterprise Institute

Training Within Industry (TWI) Templates and Downloads

8-step TWI problem solving card - as presented by IBM

• 8-steps to problem solving handy pocket card printable.

Solving Problems with TWI

• Solving problems with TWI deployment graphic.

Template of Job Breakdown Sheet

• Job breakdown sheets are created to list the steps and highlight the main factors or key points that go into completing a job.
• It also provides reasons for these key points.

TWI Job Instruction Card

• TWI Job Instruction card in a handy pocket printable.

TWI Job Methods Card

• TWI Job Methods Card in a handy pocket printable.

TWI Job Relations Card

• TWI Job Relations Card in a handy pocket printable.

Lean Solutions

James (Jim) Womack, PhD and Dan Jones

Lean Thinking, 2nd Edition

Project Management Templates

Master Schedule and Action Plan Template for One Goal (from Perfecting Patient Journeys)

• Use this template in your project tracking center so you can track both goals and action items on the same form.

End of Project Review Template (from Perfecting Patient Journeys)

• The following template will help you capture your end-of-project reflections and make decisions about what to do next.

Master Schedule Template (from Perfecting Patient Journeys)

• This template will help you answer this question by letting you include the project goals with space to indicate whether each goal is on track as originally planned and whether the scheduled progress review has taken place.

Team Board Form (from Getting the Right Things Done)

• A team board is a window on both routine and improvement work.
• The board on this template addresses both daily production and strategic issues, and is organized according to SQDCM—safety, quality, delivery, cost, and morale.

Value Proposition Template (from Perfecting Patient Journeys)

• Align the stakeholders around what will be included in addressing the problem,
• Identify the stakeholders who will be added to the project team and actively engaged in creating the current- and future-state value-stream maps,
• Identify additional stakeholders necessary to drive the implementation of the future state,
• Serve as an agreement—a proof of consensus—on the specific problem to be solved, and with the problem statement serve as authorization for the entire project.

Action Planning Template (from Perfecting Patient Journeys)

• Identify the specific changes that need to be made and translate those changes into clearly stated goals and actions (i.e., the means) to achieve those goals.
• Identify the specific methods and action steps you think will help you achieve the goals. These action steps and targets constitute the action plan to achieve a specific goal.

Value-Stream Mapping Templates

Value-stream Mapping Icons for Excel

• At the request of some of our readers we have posted the most commonly used mapping icons so that they can be downloaded for Excel spreadsheets.

Learning to See

Mike Rother and John Shook

VSM Getting Started Set

Lean Enterprise Institute , Mike Rother and John Shook

Mapping to See: Value-Stream Improvement Workshop

Beau Keyte , Jim Luckman , Kirk Paluska , Guy Parsons , John Shook , Tom Shuker and David Verble

Improvement Kata / Coaching Kata

Improvement Kata Learner's Storyboard

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