The mass of the Earth and the width of an atom are both easy to write once every number has the same shape: a number from 1 up to 10, times a power of ten.
A number in standard form is \(A \times 10^{n}\), where \(1 \le A < 10\) and \(n\) is an integer. The power \(n\) counts how many places the digits move: positive for large numbers, negative for small ones.
Fig. 1: the decimal point moves 5 places to make a number between 1 and 10, so the power is 5
Section 01
One number between 1 and 10.
Type a number into the lab and watch the decimal point hop until exactly one non-zero digit sits in front of it. Count the hops: that count is the power of ten. Then try a tiny number such as 0.000 52 and see the power turn negative.
The front number \(A\) must satisfy \(1 \le A < 10\): so \(3.4\), \(9.99\) and \(1\) are allowed, but \(34\) and \(0.34\) are not.
Large numbers (10 or more) have a positive power: \(52\,000 = 5.2 \times 10^{4}\).
Small numbers (less than 1) have a negative power: \(0.000\,52 = 5.2 \times 10^{-4}\).
Numbers from 1 to 10 have power zero: \(7.3 = 7.3 \times 10^{0}\).
Worked example Into standard form
Write (a) \(4\,060\,000\) and (b) \(0.000\,089\) in standard form.
(a) Put the point after the first non-zero digit: \(4.06\).
Count the places from \(4.06\) back to \(4\,060\,000\): the point moves 6 places, and the number is large, so the power is \(+6\).
(b) The first non-zero digit is 8, so \(A = 8.9\).
From \(0.000\,089\) to \(8.9\) the point moves 5 places right; the number is small, so the power is \(-5\).
Write \(7.25 \times 10^{-3}\) as an ordinary number.
A power of \(-3\) means divide by \(10^{3} = 1000\), so the point moves 3 places left.
\(7.25 \to 0.725 \to 0.0725 \to 0.007\,25\).
Answer: \(0.007\,25\)
Examiner trap
\(34 \times 10^{5}\) is a correct number but it is NOT in standard form, because \(34\) is not less than 10. Examiners give no mark for it. Fix it to \(3.4 \times 10^{6}\).
Section 02
Comparing and ordering.
To order numbers in standard form, compare the powers first. Only when two powers are equal do you compare the front numbers.
Numbers
Compare
Larger
\(9.1 \times 10^{3}\) and \(1.2 \times 10^{4}\)
powers 3 and 4
\(1.2 \times 10^{4}\)
\(6.5 \times 10^{-2}\) and \(8 \times 10^{-3}\)
powers \(-2\) and \(-3\): \(-2\) is bigger
\(6.5 \times 10^{-2}\)
\(2.7 \times 10^{5}\) and \(3.1 \times 10^{5}\)
same power, so 2.7 and 3.1
\(3.1 \times 10^{5}\)
Examiner trap
With negative powers, the bigger-looking power is the smaller number: \(10^{-5}\) is a hundred times SMALLER than \(10^{-3}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which of these is written in standard form?
Quick check+10 XP
Write \(37\,000\) in standard form.
Quick check+10 XP
Write \(0.0061\) in standard form.
Quick check+10 XP
Which number is the largest?
Section 03
Calculating in standard form.
For multiplying and dividing, deal with the front numbers and the powers of ten separately, using the laws of indices. For adding and subtracting, first write both numbers with the same power (or as ordinary numbers). Always tidy the answer so the front number is between 1 and 10.
Worked example Multiply without a calculator (Extended)
Work out \((6 \times 10^{4}) \times (5 \times 10^{-7})\). Give your answer in standard form.
Work out \((2 \times 10^{-3})^{2}\). Give your answer in standard form.
\(2^{2} = 4\) and \((10^{-3})^{2} = 10^{-6}\), so \(4 \times 10^{-6}\).
Examiner trap
You cannot add powers when adding numbers: \(2 \times 10^{5} + 3 \times 10^{4}\) is NOT \(5 \times 10^{9}\). Index laws are for multiplying and dividing only.
Section 04
Standard form on a calculator.
On Paper 3 and Paper 4 you may use the calculator's \(\times 10^{x}\) (or EXP) key. Read the display carefully and write the answer properly.
Key in \(3.2 \times 10^{-4}\) as 3.2, then the \(\times 10^{x}\) key, then (−) 4. Do not type × 10 as well.
A display like 1.44 08 or 1.44E8 means \(1.44 \times 10^{8}\). Write it that way; "1.44E8" scores nothing.
Round the front number as the question asks, often to 3 significant figures.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
Work out \((3 \times 10^{5}) \times (4 \times 10^{-2})\), giving your answer in standard form.
Exam style+20 XP
Work out \(\dfrac{9 \times 10^{4}}{3 \times 10^{-2}}\).
Exam style+20 XP
Work out \(5.6 \times 10^{3} - 8 \times 10^{2}\), giving your answer in standard form.
Exam style+20 XP
Light travels \(3 \times 10^{8}\) metres per second. How far does it travel in \(5 \times 10^{2}\) seconds?
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Standard form, sorted.
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