GCSE Maths

Standard Form

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Introduction Watch: Standard Form What do you mean by Standard Form Why do we use Standard Form Converting into Standard Form Addition in Standard Form Subtraction in Standard Form Multiplication in Standard Form Division in Standard Form

Introduction

Standard Form is a widely accepted way of representing mathematical expressions, numbers, or equations in a clear and structured manner.

Watch: Standard Form

What do you mean by Standard Form

Standard form is a universally recognized way of expressing mathematical concepts with clarity, precision, and consistency.

It can be written in the form:

Standard form equation showing a × 10ⁿ with arrows pointing to constant and power

where,

a is any constant which lies between 1 and 9 i.e $1 \leq a < 10$.
n can be any positive or negative whole number.

Examples:

These are some of the conversions to their respective Standard Forms.

Why do we use Standard Form

We use Standard Form because it makes numbers and equations easier to read, compare, and work with. Here’s a brief breakdown of why it’s useful:

1. Simplifies Large or Small Numbers

As we can simply write

Standard form conversions: 45,000,000 to 4.5×10⁷, 0.00098 to 9.8×10⁻⁴, 0.0000048 to 4.8×10⁻⁶

2. Comparision is Quicker

Numbers in standard form make it easier to compare magnitudes without counting zeroes.
$4.5 \times 10^{15}$ is far better and understandable than $4500000000000000$

Converting into Standard Form

There are mainly two types of numbers that can be converted in Standard Form.

Steps to convert a number to Standard Form:

1. Converting a Large Number
2. Converting a Small Number

Solved Example

Problem: Convert $0.0072$ to standard form.

SOLUTION

Place the decimal after the first non-zero digit.

The number is $0.0072$, so place the decimal after $7.2$

Count the number of places the decimal moved.

The original decimal in $0.0072$ moves $3$ places to the right. So, the exponent is $-3$.

Write the number in standard form:

Final Answer: $7.2 \times 10^{-3}$

Addition in Standard Form

Addition can be performed in Standard Form by this procedure:

Steps for Addition in Standard Form:

Make sure both of the numbers have the same power of $10$.

Adjust one number accordingly so that both exponents match.

Add the coefficients while keeping the power of $10$ the same.

Convert the result back into standard form (if necessary).

Solved Example

Problem: $4.3 \times 10^3 + 3.9 \times 10^3$

SOLUTION

Both numbers have $10^3$, so just add the coefficients:

$$4.3 + 3.9 = 7.2$$

Keep the same power of $10$:

$$7.2 \times 10^3$$

Final Answer: $7.2 \times 10^3$

Solved Example

Problem: $4.2 \times 10^5 + 5.1 \times 10^3$

SOLUTION

Convert both numbers to the same power of $10$.

$5.1 \times 10^3$ can be written as $0.051 \times 10^5$

Now add the coefficients:

$$4.2 + 0.051 = 4.251$$

Keep the power of $10$:

$$4.251 \times 10^3$$

Final Answer: $4.251 \times 10^3$

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Subtraction in Standard Form

Subtraction can also be performed in Standard Form by the given procedure:

Steps for Subtraction in Standard Form:

Make sure both of the numbers have the same power of $10$.

Adjust one number accordingly so that both exponents match.

Subtract the coefficients while keeping the power of $10$ the same.

Convert the result back into standard form (if necessary).

Solved Example

Problem: $(6.8 \times 10^3) – (2.5 \times 10^3)$

SOLUTION

Both numbers have $10^3$, so just subtract the coefficients:

$$6.8 – 2.5 = 4.3$$

Keep the same power of $10$:

$$4.3 \times 10^3$$

Final Answer: $4.3 \times 10^3$

Solved Example

Problem: $(7.5 \times 10^6) – (3.2 \times 10^4)$

SOLUTION

Convert both numbers to the same power of $10$.

$3.2 \times 10^4$ can be written as $0.032 \times 10^6$

Now subtract the coefficients:

$$7.5 – 0.032 = 7.468$$

Keep the same power of $10$:

$$7.468 \times 10^6$$

Final Answer: $7.468 \times 10^6$

Multiplication in Standard Form

Multiplication can also be performed in Standard Form by the given procedure:

Case 1: Multiplication with adjustments
Case 2: Multiplication when the coefficient is greater than 10

Solved Example

Problem: $(6.2 \times 10^3) \times (5.0 \times 10^2)$

SOLUTION

Multiply the coefficients:

$$6.2 \times 5.0 = 31.0$$

Add the exponents:

$$10^3 \times 10^2 = 10^{3+2} = 10^5$$

The coefficient is greater than $10$, so adjust:

$$31.0 \times 10^5 = 3.1 \times 10^6$$

Final Answer: $3.1 \times 10^6$

Division in Standard Form

Division can also be performed in Standard Form by the given procedure:

Case 1: Simple Division
Case 2: When the Coefficient is Less than 1

Solved Example

Problem: $(4.5 \times 10^3) \div (9.0 \times 10^5)$

SOLUTION

Divide the coefficients:

$$4.5 \div 9.0 = 0.5$$

Subtract the exponents:

$$10^3 \div 10^5 = 10^{3-5} = 10^{-2}$$

It comes out to be

$$0.5 \times 10^{-2} = 5 \times 10^{-3}$$

Final Answer: $5 \times 10^{-3}$

Solved Example

Problem: Convert $567,000,000$ to standard form.

SOLUTION

Place the decimal after the first non-zero digit:

$5.67$

Count how many places the decimal moves: $8$ places to the left

Write in standard form:

Final Answer: $5.67 \times 10^8$

Solved Example

Problem: Convert $0.000042$ to standard form.

SOLUTION

Place the decimal after the first non-zero digit:

$4.2$

Count how many places the decimal moves: $5$ places to the right

Write in standard form:

Final Answer: $4.2 \times 10^{-5}$

Solved Example

Problem: $(3.2 \times 10^4) + (4.5 \times 10^3)$

SOLUTION

Convert $4.5 \times 10^3$ to match the power of $10^4$

$$0.45 \times 10^4$$

Add the coefficients:

$$3.2 + 0.45 = 3.65$$

Keep the power of $10^4$:

$$3.65 \times 10^4$$

Final Answer: $3.65 \times 10^4$

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Standard Form

Introduction

Watch: Standard Form

What do you mean by Standard Form

Why do we use Standard Form

Converting into Standard Form

Addition in Standard Form

Need help with Standard Form?

Subtraction in Standard Form

Multiplication in Standard Form

Division in Standard Form

Ready to test your knowledge?

Want a Tutor for This Topic?

Ready to Get Started?

Struggling with a topic?

Introduction

Watch: Standard Form

What do you mean by Standard Form

Why do we use Standard Form

Converting into Standard Form

Addition in Standard Form

Need help with Standard Form?

Subtraction in Standard Form

Multiplication in Standard Form

Division in Standard Form

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