Standard Form

This lesson covers writing numbers in standard form (also called scientific notation), converting between standard form and ordinary numbers, performing calculations in standard form, and ordering numbers given in standard form. Standard form is a key Number topic in the Edexcel GCSE Mathematics (1MA1) specification.

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Key Vocabulary

Term	Meaning
Standard form	A way of writing very large or very small numbers as $A \times 10^n$A×10n
Ordinary number	A number written in the normal way, e.g. 45,000
Significant figures	The meaningful digits in a number

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What Is Standard Form?

A number in standard form is written as:

$A \times 10^n$A×10n

where:

• $1 \leq A < 10$1≤A<10 (A is at least 1 but less than 10)
• n is an integer (positive, negative or zero)

Examples

Ordinary number	Standard form
45,000	$4.5 \times 10^{4}$4.5×104
3,200,000	$3.2 \times 10^{6}$3.2×106
0.00072	$7.2 \times 10^{-4}$7.2×10−4
0.0000009	$9 \times 10^{-7}$9×10−7
8.1	$8.1 \times 10^{0}$8.1×100

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Converting from Ordinary Numbers to Standard Form

Large Numbers (n is positive)

Move the decimal point to the left until you have a number between 1 and 10. The number of places you moved is the power of 10.

Worked Example 1

Write 6,730,000 in standard form.

$6,730,000 \to move$6,730,000→move the decimal point 6 places to the left $\to 6.73$→6.73

Answer: $6.73 \times 10^{6}$6.73×106

Small Numbers (n is negative)

Move the decimal point to the right until you have a number between 1 and 10. The number of places you moved gives a negative power.

Worked Example 2

Write 0.000348 in standard form.

$0.000348 \to move$0.000348→move the decimal point 4 places to the right $\to 3.48$→3.48

Answer: $3.48 \times 10^{-4}$3.48×10−4

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Converting from Standard Form to Ordinary Numbers

Positive Power

Move the decimal point to the right.

Worked Example 3

Write $2.57 \times 10^{5}$2.57×105 as an ordinary number.

Move decimal 5 places right: 257,000

Answer: 257,000

Negative Power

Move the decimal point to the left.

Worked Example 4

Write $8.1 \times 10^{-3}$8.1×10−3 as an ordinary number.

Move decimal 3 places left: 0.0081

Answer: 0.0081

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Calculating with Standard Form

Multiplication

Multiply the A values and add the powers of 10.

Worked Example 5

Calculate $(3 \times 10^{4}) \times (5 \times 10^{3})$(3×104)×(5×103). Give your answer in standard form.

• Multiply the A values: $3 \times 5 = 15$3×5=15
• Add the powers: $10^{4} \times 10^{3} = 10^{7}$104×103=107
• Combined: $15 \times 10^{7}$15×107

But 15 is not between 1 and 10, so adjust:

• $15 = 1.5 \times 10^{1}$15=1.5×101
• $1.5 \times 10^{1} \times 10^{7} = 1.5 \times 10^{8}$1.5×101×107=1.5×108

Answer: $1.5 \times 10^{8}$1.5×108

Division

Divide the A values and subtract the powers of 10.

Worked Example 6

Calculate $(8.4 \times 10^{6}) \div (2.1 \times 10^{2})$(8.4×106)÷(2.1×102). Give your answer in standard form.

• Divide the A values: $8.4 \div 2.1 = 4$8.4÷2.1=4
• Subtract the powers: $10^{6} \div 10^{2} = 10^{4}$106÷102=104

Answer: $4 \times 10^{4}$4×104

Worked Example 7

Calculate $(2.4 \times 10^{-3}) \div (8 \times 10^{2})$(2.4×10−3)÷(8×102). Give your answer in standard form.

• $2.4 \div 8 = 0.3$2.4÷8=0.3
• $10^{-3} \div 10^{2} = 10^{-5}$10−3÷102=10−5
• Combined: $0.3 \times 10^{-5}$0.3×10−5

Adjust: $0.3 = 3 \times 10^{-1}$0.3=3×10−1, so $3 \times 10^{-1} \times 10^{-5} = 3 \times 10^{-6}$3×10−1×10−5=3×10−6

Answer: $3 \times 10^{-6}$3×10−6

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Addition and Subtraction in Standard Form

For addition and subtraction, you must convert to the same power of 10 (or convert to ordinary numbers).

Worked Example 8

Calculate $(4.5 \times 10^{5}) + (3.2 \times 10^{4})$(4.5×105)+(3.2×104).

Method 1 — Convert to ordinary numbers:

• $4.5 \times 10^{5} = 450,000$4.5×105=450,000
• $3.2 \times 10^{4} = 32,000$3.2×104=32,000
• Sum $= 482,000 = 4.82 \times 10^{5}$=482,000=4.82×105

Method 2 — Make powers the same:

• $3.2 \times 10^{4} = 0.32 \times 10^{5}$3.2×104=0.32×105
• $4.5 \times 10^{5} + 0.32 \times 10^{5} = 4.82 \times 10^{5}$4.5×105+0.32×105=4.82×105

Answer: $4.82 \times 10^{5}$4.82×105

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Ordering Numbers in Standard Form

Worked Example 9

Put these in ascending order: $3.1 \times 10^{-2}$3.1×10−2, $4.7 \times 10^{-3}$4.7×10−3, $2.9 \times 10^{-2}$2.9×10−2, $5.6 \times 10^{-4}$5.6×10−4

Step 1: Compare powers of 10 first. Larger (less negative) powers mean larger numbers.

• $10^{-4}$10−4 is smallest, then $10^{-3}$10−3, then $10^{-2}$10−2

Step 2: For numbers with the same power, compare A values.

Ascending order: $5.6 \times 10^{-4}$5.6×10−4, $4.7 \times 10^{-3}$4.7×10−3, $2.9 \times 10^{-2}$2.9×10−2, $3.1 \times 10^{-2}$3.1×10−2

Edexcel Exam Tip: When ordering standard form numbers, look at the power of 10 FIRST. Only compare the A values when the powers are the same. This is a common 2-mark question on Edexcel papers.

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Standard Form on a Calculator

On most scientific calculators:

• Use the $\times10^x$×10x or EXP button to enter standard form.
• To enter $3.2 \times 10^{5}$3.2×105: press 3.2, then $\times10^x ($×10x(or EXP), then 5.
• The display may show 3.2E5 or $3.2^{05}$3.205.

Important: Do NOT type "$\times 10 ^ 5$×105" — use the dedicated EXP or $\times10^x$×10x button.

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Common Mistakes and Misconceptions

Mistake	Correction
Writing $34.5 \times 10^{3}$34.5×103 in standard form	A must be between 1 and 10: $3.45 \times 10^{4}$3.45×104
Writing $0.7 \times 10^{5}$0.7×105 in standard form	A must be $\geq 1$≥1: $7 \times 10^{4}$7×104
Adding powers when adding standard form numbers	You can only add powers when MULTIPLYING
Forgetting to adjust the answer into standard form	Always check that $1 \leq A < 10$1≤A<10
Confusing $10^{-3} (= 0.001)$10−3(=0.001) with $-10^{3} (= -1000)$−103(=−1000)	$10^{-3}$10−3 is a very small POSITIVE number

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Practice Problems

1. Write 92,500,000 in standard form.
2. Write 0.0000067 in standard form.
3. Write $4.03 \times 10^{-5}$4.03×10−5 as an ordinary number.
4. Calculate $(6 \times 10^{3}) \times (4 \times 10^{5})$(6×103)×(4×105), giving your answer in standard form.
5. Calculate $(9.6 \times 10^{8}) \div (1.2 \times 10^{3})$(9.6×108)÷(1.2×103), giving your answer in standard form.
6. Calculate $(5.7 \times 10^{4}) + (8.3 \times 10^{3})$(5.7×104)+(8.3×103), giving your answer in standard form.
7. Order from smallest to largest: $6 \times 10^{3}$6×103, $7 \times 10^{2}$7×102, $4.5 \times 10^{3}$4.5×103, $9.8 \times 10^{2}$9.8×102
8. The distance from the Earth to the Sun is approximately $1.5 \times 10^{8} km$1.5×108km. Light travels at approximately $3 \times 10^{5} km$3×105km per second. How many seconds does light take to travel from the Sun to the Earth?

Answers