Standard Form (Scientific Notation)

Standard form, also called scientific notation, is the method used in GCSE and IGCSE Maths to write very large and very small numbers in a compact, consistent way. Every number is written as a × 10n, where 1 ≤ a < 10 and n is an integer. On this page you will learn how to convert between ordinary numbers and standard form, how to add, subtract, multiply and divide numbers in standard form, and how to avoid the mistakes that cost marks in exams. Work through the step-by-step examples, then test yourself with the randomly generated, auto-marked practice questions below.

Prior Knowledge You should be comfortable with place value and powers of ten, and with the laws of indices, since multiplying and dividing in standard form means adding and subtracting the powers.
⇩ Jump to Practice Questions ⇩

Watch the Decimal Move

Large number → move decimal LEFT

Small number → move decimal RIGHT

The arrow bounces once for each place the decimal moves. Left moves give \(n>0\); right moves give \(n<0\).

What Is Standard Form?

A number is in standard form if it is written as \(a \times 10^n\) where \(1 \le a < 10\) and \(n\) is an integer.

How to Convert (2 Steps)

  1. Find \(a\): place the decimal after the first non-zero digit so that \(1 \le a < 10\).
  2. Find \(n\): count how many places the decimal moved. Left gives \(n > 0\); right gives \(n < 0\).

Core Ideas

The form
Always \(a \times 10^n\) with \(1 \le a < 10\) and \(n\) an integer.
Big vs small
Large numbers have \(n > 0\). Small decimals have \(n < 0\).
Multiply / Divide
Multiply or divide the \(a\)s; add or subtract the powers.
Add / Subtract
Match the powers of ten first, then combine the \(a\)s.

Converting to Standard Form

💡 Large Number to Standard Form

Write \(5\,600\,000\) in standard form.

1
Find \(a\) Place the decimal after the first non-zero digit.
\[5\,600\,000 \longrightarrow a = 5.6\]
2
Find \(n\) The decimal moved 6 places to the left, so \(n = 6\).
3
Write the answer
\[5\,600\,000 = 5.6 \times 10^6\]

💡 Small Number to Standard Form

Write \(0.000032\) in standard form.

1
Find \(a\) Place the decimal after the first non-zero digit.
\[0.000032 \longrightarrow a = 3.2\]
2
Find \(n\) The decimal moved 5 places to the right, so \(n = -5\).
3
Write the answer
\[0.000032 = 3.2 \times 10^{-5}\]

💡 Standard Form to Ordinary Number (Large)

Write \(4.5 \times 10^{3}\) as an ordinary number.

1
Read the power \(n = 3\) is positive, so move the decimal 3 places to the right.
2
Write the answer
\[4.5 \times 10^{3} = 4500\]

💡 Standard Form to Ordinary Number (Small)

Write \(9.1 \times 10^{-3}\) as an ordinary number.

1
Read the power \(n = -3\) is negative, so move the decimal 3 places to the left.
2
Write the answer
\[9.1 \times 10^{-3} = 0.0091\]

Adding & Subtracting

💡 Adding: Powers Already Match

Calculate \((3.1 \times 10^4) + (2.5 \times 10^4)\).

1
Check the powers Both terms have \(10^4\). Powers match, so no rewriting is needed.
2
Add the coefficients
\[(3.1 + 2.5) \times 10^4 = 5.6 \times 10^4\]

💡 Adding: Powers Don't Match

Calculate \((3.6 \times 10^3) + (7.2 \times 10^4)\).

1
Match the powers Rewrite the \(10^3\) term as \(10^4\):
\[3.6 \times 10^3 = 0.36 \times 10^4\]
2
Add the coefficients
\[(0.36 + 7.2) \times 10^4 = 7.56 \times 10^4\]

💡 Subtracting: Powers Already Match

Calculate \((9.8 \times 10^5) - (3.2 \times 10^5)\).

1
Check the powers Both terms have \(10^5\). Powers match.
2
Subtract the coefficients
\[(9.8 - 3.2) \times 10^5 = 6.6 \times 10^5\]

💡 Subtracting: Powers Don't Match

Calculate \((9.1 \times 10^6) - (4.3 \times 10^5)\).

1
Match the powers Rewrite the \(10^5\) term as \(10^6\):
\[4.3 \times 10^5 = 0.43 \times 10^6\]
2
Subtract the coefficients
\[(9.1 - 0.43) \times 10^6 = 8.67 \times 10^6\]

Multiplying & Dividing

💡 Multiplying

Calculate \((2 \times 10^3) \times (3 \times 10^5)\).

1
Multiply the coefficients
\[2 \times 3 = 6\]
2
Add the powers
\[\begin{aligned} 10^3 \times 10^5 &= 10^{3+5} \\ &= 10^8 \end{aligned}\]
3
Write the answer
\[(2 \times 10^3) \times (3 \times 10^5) = 6 \times 10^8\]

💡 Dividing

Calculate \((6 \times 10^{7}) \div (3 \times 10^{4})\).

1
Divide the coefficients
\[6 \div 3 = 2\]
2
Subtract the powers
\[\begin{aligned} 10^7 \div 10^4 &= 10^{7-4} \\ &= 10^3 \end{aligned}\]
3
Write the answer
\[(6 \times 10^7) \div (3 \times 10^4) = 2 \times 10^3\]

💡 Dividing: Rounding to 3 Significant Figures

Calculate \((5 \times 10^{7}) \div (3 \times 10^{2})\). Give your answer to 3 significant figures.

1
Divide the coefficients
\[5 \div 3 = 1.6667\ldots\]
2
Subtract the powers
\[10^7 \div 10^2 = 10^5\]
3
Combine, then round the coefficient to 3 s.f.
\[\begin{aligned} (5 \times 10^7) \div (3 \times 10^2) &= 1.6667\ldots \times 10^5 \\ &= 1.67 \times 10^5 \text{ (3 s.f.)} \end{aligned}\]

Where You Will Use Standard Form

🔭
Astronomy
Distance to the Sun
1.5 × 1011 m
⚡
Physics
Mass of an electron
9.1 × 10−31 kg
🔬
Biology
Width of a bacterium
2.0 × 10−6 m
💻
Computing
Transistors on a chip
5.0 × 1010
🌍
Geography
World population
8.1 × 109

🔑 Key Points

  • Standard form is \(a \times 10^n\) with \(1 \le a < 10\) and integer \(n\).
  • Left moves give \(n > 0\); right moves give \(n < 0\).
  • For \(+\) or \(-\): match powers first, then combine the \(a\)s.
  • For \(\times\) or \(\div\): multiply/divide the \(a\)s; add/subtract the powers.
  • If a multiply or divide does not come out exactly, round the coefficient to 3 significant figures.

⚠️ Common Pitfalls

  • \(a\) must satisfy \(1 \le a < 10\). Rewrite \(12.6 \times 10^3\) as \(1.26 \times 10^4\).
  • Never add or subtract until the powers match.
  • Negative powers give small decimals, not negative numbers.
⇩ Jump to Practice Questions ⇩

Standard form runs on the laws of indices (multiplying and dividing add and subtract the powers), and awkward answers are rounded to 3 significant figures. Brush up on index laws and rounding to significant figures next.

Next: Laws of Indices →

Standard Form: Practice Room

Practise converting between standard form and ordinary numbers, and performing addition, subtraction, multiplication and division with powers of ten. Every room generates new questions automatically and marks your answers instantly. For standard form answers use the format 3.2x10^5 or 3.2x10^-4; if a division does not come out exactly, round the coefficient to 3 significant figures.

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Standard form answers: type 3.2x10^5 or 3.2x10^-4. Ordinary number answers: write the full number, no commas (e.g. 45000 or 0.0091).

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