Standard Form
Standard form, also called standard index form, writes any positive number as A x 10^n, where A is between 1 and 10 and n is an integer power of 10. It is used to write very large numbers, such as populations or distances in space, or very small numbers, such as the mass of a bacterium, compactly, and to compare the size of numbers that would otherwise have long strings of zeros.
Before you start
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Method
- To convert an ordinary number into standard form, place the decimal point so there is exactly one non-zero digit before it, giving the front number A.
- Count how many places the decimal point moved to reach that position - this count becomes the power n.
- If the original number was 10 or more, n is positive; if the original number was between 0 and 1, n is negative.
- To convert from standard form back to an ordinary number, move the decimal point the number of places shown by n (right for positive n, left for negative n), filling any gaps with zeros.
- To order numbers given in standard form, compare the powers of 10 first; only compare the front numbers A when two powers are equal.
- To multiply numbers in standard form, multiply the front numbers together and add the powers of 10, then adjust the result back into standard form if the front number is no longer between 1 and 10.
Worked example
Write 7,400,000 in standard form. Then order 7.4 x 10^6, 6.9 x 10^7 and 8.1 x 10^5 from smallest to largest.
- For 7,400,000, place the decimal point after the first non-zero digit (7) to get the front number 7.4.
- Count the places moved: from 7,400,000 to 7.4 the point moved 6 places, so the power is 6, giving 7.4 x 10^6.
- To order the three numbers, compare their powers of 10 first: 10^5, 10^6 and 10^7.
- The smallest power gives the smallest number, so 8.1 x 10^5 is smallest, then 7.4 x 10^6, then 6.9 x 10^7 is largest.
- Smallest to largest: 8.1 x 10^5, 7.4 x 10^6, 6.9 x 10^7.
Practice questions
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Q1Write 45,000 in standard form.Show answer
Answer: 4.5 x 10^4
Q2Write 0.00062 in standard form.Show answer
Answer: 6.2 x 10^-4
Q3Write 3.05 x 10^5 as an ordinary number.Show answer
Answer: 305,000
Q4Write 9 x 10^-3 as an ordinary number.Show answer
Answer: 0.009
Q5Put these numbers in order, smallest first: 5.2 x 10^3, 4.9 x 10^4, 5.2 x 10^2.Show answer
Answer: 5.2 x 10^2, 5.2 x 10^3, 4.9 x 10^4
Q6Work out (3 x 10^6) x (2.5 x 10^3), giving your answer in standard form.Show answer
Answer: 7.5 x 10^9
Q7Work out (6 x 10^4) x (5 x 10^2), giving your answer in standard form.Show answer
Answer: 3 x 10^7 (the front numbers give 30, which must be adjusted to 3 x 10^1)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The population of the UK is approximately 6.8 x 10^7. The population of Wales is approximately 3.1 x 10^6. (a) Write both numbers as ordinary numbers. (b) Work out approximately how many times larger the UK population is than the population of Wales, giving your answer to the nearest whole number.
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A single bacterium has a mass of 9.5 x 10^-13 grams. Work out the total mass, in grams, of 4 x 10^5 bacteria. Give your answer in standard form.
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A teacher writes four numbers on the board: 2.6 x 10^4, 2.6 x 10^5, 2.6 x 10^3 and 26000. (a) Which two of these numbers are equal in value? (b) Write the largest of the four numbers as an ordinary number.
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Free printable worksheet
Want more practice on paper? Download the standard form worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 13 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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