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Standard Form

What you'll learn

  • Recognise what standard form looks like.
  • Convert very large and very small numbers into and out of standard form.
  • Multiply, divide, add, subtract, and compare numbers involving powers of 10.
  • Use standard form in worded questions, including distance, mass, and formula problems.

Ordinary numbers and powers of 10

Before standard form, you need to be confident with place value and powers of 10.

Definition

Ordinary number and power of 10

  • An ordinary number is a number written in the usual way, like 42000 or 0.0036.
  • A power of 10 is a number such as 10310^3103 or 10−210^{-2}10−2.
  • The small raised number is called the exponent or index. It tells you how many places the decimal point moves.

A positive exponent makes the number bigger. A negative exponent makes the number smaller.

Key Idea

Decimal movement

Multiplying by 10n10^n10n moves the decimal point right if nnn is positive, and left if nnn is negative.

Example

Using powers of 10

  1. For 7.41×1037.41 \times 10^37.41×103, the exponent is positive 3, so move the decimal point 3 places to the right.

A place-value arrow shows that a positive power of 10 moves the decimal point to the right.

  1. This gives 7410.

  2. For 7.41×10−27.41 \times 10^{-2}7.41×10−2, the exponent is negative 2, so move the decimal point 2 places to the left.

A negative power of 10 moves the decimal point to the left, adding a zero placeholder if needed.

  1. This gives 0.0741.

What counts as standard form?

Definition

Standard form

A number is in standard form when it is written as a×10na \times 10^na×10n, where 1≤a<101 \le a < 101≤a<10 and nnn is an integer, meaning a whole number. The front number aaa is called the coefficient.

Standard form means the coefficient is in the interval from 1 up to but not including 10.

So the coefficient must be at least 1 and less than 10.

For example, 4.6×1084.6 \times 10^84.6×108 is in standard form, but 46×10746 \times 10^746×107 is not.

Example

Fixing a number that is nearly standard form

  1. Write 0.48×1050.48 \times 10^50.48×105 in standard form.

The coefficient 0.48 is adjusted into the valid range by moving the decimal point one place right and reducing the power of 10 by 1.

  1. The coefficient 0.48 is too small, because it is less than 1.

  2. Rewrite 0.48 as 4.8×10−14.8 \times 10^{-1}4.8×10−1.

  3. Combine the powers of 10: 10−1×105=10410^{-1} \times 10^5 = 10^410−1×105=104.

  4. So 0.48×105=4.8×1040.48 \times 10^5 = 4.8 \times 10^40.48×105=4.8×104.

Common Mistake

Forgetting the front-number rule

12.3×10412.3 \times 10^412.3×104 is not finished because 12.3 is not less than 10. Rewrite it as 1.23×1051.23 \times 10^51.23×105.

Converting from standard form to ordinary numbers

To turn standard form into an ordinary number, move the decimal point.

  • Positive exponent: move right.
  • Negative exponent: move left.
  • Add zeros if you need extra place holders.
Example

Standard form to ordinary numbers

  1. Write 8.17×1048.17 \times 10^48.17×104 as an ordinary number.

Moving the decimal point four places right converts 8.17 \times 10^4 into an ordinary number.

  1. The exponent is positive 4, so move the decimal point 4 places right: 8.17→817008.17 \to 817008.17→81700.

  2. So 8.17×104=817008.17 \times 10^4 = 817008.17×104=81700.

  3. Write 5.09×10−35.09 \times 10^{-3}5.09×10−3 as an ordinary number.

  4. The exponent is negative 3, so move the decimal point 3 places left: 5.09→0.005095.09 \to 0.005095.09→0.00509.

  5. So 5.09×10−3=0.005095.09 \times 10^{-3} = 0.005095.09×10−3=0.00509.

Tip

Count places, not zeros

When converting, count the number of decimal-place moves. Do not just count how many zeros appear.

Converting ordinary numbers to standard form

To convert an ordinary number into standard form:

  1. Put the decimal point after the first non-zero digit.
  2. Count how many places the decimal point moved.
  3. Use a positive exponent for large numbers.
  4. Use a negative exponent for small decimals between 0 and 1.
Example

Large ordinary number to standard form

  1. Write 64 800 000 in standard form.

For a large ordinary number, the decimal point moves left until the coefficient is between 1 and 10.

  1. The first non-zero digit is 6, so the coefficient will be 6.48.

  2. The decimal point has moved 7 places left from 64800000 to 6.48.

  3. Because the original number was large, the exponent is positive.

  4. So 64 800 000 is 6.48×1076.48 \times 10^76.48×107.

Example

Small decimal to standard form

  1. Write 0.000731 in standard form.

For a small decimal, the decimal point moves right to make the coefficient, so the exponent is negative.

  1. The first non-zero digit is 7, so the coefficient will be 7.31.

  2. The decimal point has moved 4 places right to make 7.31.

  3. Because the original number was small, the exponent is negative.

  4. So 0.000731 is 7.31×10−47.31 \times 10^{-4}7.31×10−4.

Common Mistake

Wrong sign on the exponent

Large ordinary numbers have positive exponents. Small decimals between 0 and 1 have negative exponents.

Multiplying and dividing in standard form

When multiplying:

  • Multiply the coefficients.
  • Add the exponents.

When dividing:

  • Divide the coefficients.
  • Subtract the exponents.
(a×10m)(b×10n)=ab×10m+n\left(a \times 10^m\right)\left(b \times 10^n\right)=ab \times 10^{m+n}(a×10m)(b×10n)=ab×10m+n (a×10m)÷(b×10n)=ab×10m−n\left(a \times 10^m\right)\div\left(b \times 10^n\right)=\frac{a}{b}\times 10^{m-n}(a×10m)÷(b×10n)=ba​×10m−n
Example

Multiplying two standard form numbers

  1. Work out (7×105)(3×10−3)\left(7 \times 10^5\right)\left(3 \times 10^{-3}\right)(7×105)(3×10−3).

Multiplying in standard form means multiplying coefficients and adding exponents, then adjusting the coefficient if necessary.

  1. Multiply the coefficients: 7 multiplied by 3 is 21.

  2. Add the exponents: 5 + (-3) = 2, giving 21×10221 \times 10^221×102.

  3. The coefficient 21 is too big for standard form.

  4. Rewrite 21×10221 \times 10^221×102 as 2.1×1032.1 \times 10^32.1×103.

Example

Dividing two standard form numbers

  1. Work out (8.4×10−5)÷(2.1×10−7)\left(8.4 \times 10^{-5}\right)\div\left(2.1 \times 10^{-7}\right)(8.4×10−5)÷(2.1×10−7).

  2. Divide the coefficients: 8.4 divided by 2.1 is 4.

  3. Subtract the exponents: -5 - (-7) = 2.

  4. So the answer is 4×1024 \times 10^24×102.

Common Mistake

Subtracting a negative exponent

Be careful with two minus signs. For example, -5 - (-7) means -5 + 7, which gives 2.

Fractions involving decimals

Sometimes it is easier to convert decimals into standard form first, especially when the numbers are very small.

Example

A fraction with small decimals

  1. Work out 0.06×0.0040.0008\frac{0.06 \times 0.004}{0.0008}0.00080.06×0.004​ in standard form.

  2. Convert each decimal: 0.06=6×10−20.06 = 6 \times 10^{-2}0.06=6×10−2, 0.004=4×10−30.004 = 4 \times 10^{-3}0.004=4×10−3, and 0.0008=8×10−40.0008 = 8 \times 10^{-4}0.0008=8×10−4.

  3. Rewrite the calculation:

    (6×10−2)(4×10−3)8×10−4\frac{\left(6 \times 10^{-2}\right)\left(4 \times 10^{-3}\right)}{8 \times 10^{-4}}8×10−4(6×10−2)(4×10−3)​

  4. Multiply the top: 6 multiplied by 4 is 24, and -2 + (-3) = -5, so the top is 24×10−524 \times 10^{-5}24×10−5.

  5. Divide by the bottom: 24 divided by 8 is 3, and -5 - (-4) = -1.

  6. The answer is 3×10−13 \times 10^{-1}3×10−1.

Adding, subtracting, and ordering

For addition and subtraction, do not just add or subtract the exponents. Either convert to ordinary numbers, or rewrite the numbers using the same power of 10.

For Grade 5 questions, converting to ordinary numbers is often the safest method.

Example

Finding a difference

  1. A large moon has diameter 6.0×1046.0 \times 10^46.0×104 km. A smaller moon has diameter 7.5×1037.5 \times 10^37.5×103 km.

  2. Convert both to ordinary numbers: 6.0×104=600006.0 \times 10^4 = 600006.0×104=60000 and 7.5×103=75007.5 \times 10^3 = 75007.5×103=7500.

  3. Subtract: 60000 - 7500 = 52500.

  4. Convert 52500 into standard form.

  5. The difference is 5.25×1045.25 \times 10^45.25×104 km.

Example

Ordering numbers

  1. Convert each number to an ordinary number:

    • 4.8×102=4804.8 \times 10^2 = 4804.8×102=480
    • 0.048×102=4.80.048 \times 10^2 = 4.80.048×102=4.8
    • 4800×10−4=0.484800 \times 10^{-4} = 0.484800×10−4=0.48
    • 48 stays as 48

  2. Order the ordinary numbers from smallest to largest: 0.48, 4.8, 48, 480.

  3. So the original numbers in order are 4800×10−44800 \times 10^{-4}4800×10−4, 0.048×1020.048 \times 10^20.048×102, 48, 4.8×1024.8 \times 10^24.8×102.

Common Mistake

Adding the powers

For addition or subtraction, do not add the exponents. 2×105+3×105=5×1052 \times 10^5 + 3 \times 10^5 = 5 \times 10^52×105+3×105=5×105, but different powers need careful converting first.

Worded problems and formula questions

In worded questions, first decide the operation.

  • “Total” usually means add.
  • “Difference” means subtract.
  • “Times larger” means divide.
  • “12 objects each weighing...” means multiply.
  • Time is distance divided by speed.
Example

A distance and speed question

  1. A signal travels 84 million km at a speed of 2.8×1052.8 \times 10^52.8×105 km/s.

  2. Write 84 million in standard form: 8.4×1078.4 \times 10^78.4×107 km.

  3. Time is distance divided by speed.

  4. Calculate:

    8.4×1072.8×105=3×102\frac{8.4 \times 10^7}{2.8 \times 10^5}=3 \times 10^22.8×1058.4×107​=3×102

  5. The time is 3×1023 \times 10^23×102 seconds.

Example

Using a sphere formula

  1. A sphere has radius 2.0×1032.0 \times 10^32.0×103 metres. Use V=43πr3V=\frac{4}{3}\pi r^3V=34​πr3.

  2. Cube the radius:

    (2.0×103)3=8.0×109\left(2.0 \times 10^3\right)^3=8.0 \times 10^9(2.0×103)3=8.0×109

  3. Multiply by 43π\frac{4}{3}\pi34​π: using a calculator, V≈3.351×1010V \approx 3.351 \times 10^{10}V≈3.351×1010.

  4. Rounded to 1 decimal place in standard form, the volume is 3.4×10103.4 \times 10^{10}3.4×1010 cubic metres.

Tip

Calculator display

Some calculators show 1.2E51.2\text{E}51.2E5 to mean 1.2×1051.2 \times 10^51.2×105. Always rewrite it clearly in standard form in your answer.

Exam technique

In the exam

  1. Check the coefficient: it must be at least 1 and less than 10.

  2. For multiplication and division, handle the coefficients and exponents separately, then adjust the final answer into standard form.

  3. In worded questions, identify the operation before calculating: total, difference, times larger, or distance divided by speed.

Self review

Check yourself

  • Is 0.72×1050.72 \times 10^50.72×105 in standard form? If not, what should it become?

  • When you divide 10−310^{-3}10−3 by 10−710^{-7}10−7, what happens to the exponents?

  • How would you write 56 million in standard form?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Standard Form Revision Guide

  1. GCSE
  2. /Maths
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