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Standard Form

Standard Form

Writing numbers in standard form

Arranging Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. What standard form is
  4. Writing a number in standard form
  5. Back to an ordinary number
  6. Fixing numbers that are not in standard form
  7. Ordering numbers in standard form
  8. Ways they test it
  9. Mistakes to avoid
  10. Summary
  11. Practice
  12. Can I do this?
  13. Quiz

In this note

Syllabus 1.8: standard form
  • 1.8Know what standard form is: \(A\times10^n\) where \(1\le A<10\) and \(n\) is a whole number (positive or negative).
  • 1.8Convert large and small numbers into standard form, and out of it back to ordinary numbers.
  • 1.8Correct numbers that are NOT in standard form (like \(45\times10^5\)); order them and round them.

Helpful first: negative indices. Next: calculating in standard form.

Formula bank

Standard formLearn it\(A\times10^{n}\), \(1\le A<10\), n an integerone non-zero digit before the point.Use it when very big or very small numbers.
Sign of the powerLearn itnumber \(\ge10\): \(n>0\); number \(<1\): \(n<0\); 1 to 9.99…: \(n=0\)Use it when every conversion.
Fixing ALearn itA ÷ 10 → \(n+1\); A × 10 → \(n-1\)\(45\times10^5=4.5\times10^6\).Use it when A not between 1 and 10.

What standard form is

The Sun is about 150 000 000 km away; a dust particle has a mass of about 0.000 000 000 7 kg. Standard form writes such numbers as a number from 1 to under 10, times a power of 10: \(1.5\times10^{8}\) and \(7\times10^{-10}\). The power counts how many places the decimal point moves.

Big numberSmall number
Ordinary372 000 0000.000 257
Standard form\(3.72\times10^{8}\)\(2.57\times10^{-4}\)
Power of 10positivenegative

Writing a number in standard form

General methodHow to write any number in standard form

  1. A: put the decimal point just after the FIRST non-zero digit (372 000 000 → 3.72; 0.000 257 → 2.57). Drop end zeros; keep middle zeros.
  2. Count how many places the point moves from A back to the original number.
  3. Original 10 or bigger → positive power. Less than 1 → negative power.
  4. Check \(1\le A<10\) and that it multiplies back to the original.
Counting decimal places3720000003.72 → 372 000 000: 8 jumps right → ×10⁸00002572.57 → 0.000 257: 4 jumps left → ×10⁻⁴
Right (bigger) = positive power; left (smaller) = negative power.

Shortcut for small numbers: the power is MINUS the decimal place of the first non-zero digit (0.007 09: the 7 is 3rd → \(10^{-3}\)).

Example Write 0.007 09 in standard form.

The 7 is in the 3rd decimal place; keep the middle 0.

  1. \(A=7.09\)

Answer \(7.09\times10^{-3}\)

Try it yourself

1
Write 66 000 in standard form.
Show answer

Answer \(6.6\times10^{4}\)

2
Write 52 million in standard form.
Show answer

Answer \(5.2\times10^{7}\)

3
Write 0.000 052 3 in standard form.
Show answer

Answer \(5.23\times10^{-5}\)

Back to an ordinary number

General methodHow to write a standard form number as an ordinary number

  1. Positive power: move the point that many places RIGHT, filling with zeros.
  2. Negative power: move it that many places LEFT. There is one zero fewer after the point than the power (10⁻⁴ → three zeros).
Standard formMove the pointOrdinary number
\(4.06\times10^{5}\)5 places right406 000
\(1.23\times10^{-4}\)4 places left0.000 123
\(6.05\times10^{-2}\)2 places left0.0605

Try it yourself

1
Write \(7.54\times10^{-4}\) as an ordinary number.
Show answer

Answer 0.000 754

2
Write \(3.05\times10^{4}\) as an ordinary number.
Show answer

Answer 30 500

Fixing numbers that are not in standard form

General methodHow to fix A when it is not between 1 and 10

  1. A too big: divide A by 10 (or 100, …) and ADD that many to the power: \(45\times10^5=4.5\times10^{6}\).
  2. A too small: multiply A by 10 and SUBTRACT 1 from the power: \(0.6\times10^{-3}=6\times10^{-4}\).
  3. Ordinary numbers times powers: convert each, then add the powers: \(4500\times1000^2=4.5\times10^3\times10^6=4.5\times10^{9}\).

Try it yourself

1
Write \(23\times10^{-5}\) in standard form.
Show answer

Answer \(2.3\times10^{-4}\)

2
Write \(0.08\times10^{9}\) in standard form.
Show answer

Answer \(8\times10^{7}\)

Ordering numbers in standard form

General methodHow to order numbers in standard form

  1. Put every number in standard form (ordinary numbers and fractions too).
  2. Compare the powers first; with negatives, \(10^{-3}\) is bigger than \(10^{-4}\).
  3. Same power? Compare the A values.
  4. Write the answer using the numbers as they were given.

Example Order, smallest first: \(2.1\times10^{-3}\), \(4.2\times10^{-4}\), \(1.7\times10^{-5}\), \(3.5\times10^{-4}\).

Powers −5, −4, −4, −3.

  1. −4s: 3.5 before 4.2

Answer \(1.7\times10^{-5},\ 3.5\times10^{-4},\ 4.2\times10^{-4},\ 2.1\times10^{-3}\)

Ways they test it

Type 1 Into / out of standard form

"Write 8 600 000 000 in standard form." / "Write 9 × 10³ as an ordinary number." (1 mark each)

Spot it the words "standard form".

  1. Count the jumps.
  2. Check the sign of the power.

For example 0.017

  1. 1 is in the 2nd decimal place

Answer \(1.7\times10^{-2}\)

Type 2 Fix the form

"Write \(23\times10^{-5}\) in standard form."

Spot it A outside 1 to 10.

  1. Opposite change to the power.

Type 3 Order

"Write in order of size, smallest first." (1-2 marks)

Spot it a mixed list.

  1. Standard form for all.
  2. Powers, then A.

Type 4 Round in standard form

"…give your answer in standard form correct to 3 s.f."

Spot it a calculator answer.

  1. Round A, then write × 10ⁿ.

Mistakes to avoid

  • ✗ \(37.2\times10^{7}\)✓ \(3.72\times10^{8}\)A must be less than 10.
  • ✗ 0.000 257 = \(2.57\times10^{4}\)✓ \(2.57\times10^{-4}\)Less than 1 → negative power.
  • ✗ \(1.23\times10^{-4}=0.0000123\)✓ 0.000 123One zero fewer than the power.
  • ✗ 15 060 = \(1.56\times10^4\)✓ \(1.506\times10^{4}\)Keep middle zeros.
  • ✗ 2.5E-07✓ \(2.5\times10^{-7}\)Calculator notation is not accepted.

Summary

  • \(A\times10^n\) with \(1\le A<10\), n an integer.
  • n = the number of jumps of the decimal point; ≥ 10 positive, < 1 negative.
  • Fix a wrong A by moving the power the opposite way.
  • Order by the power first, then by A.

If you see …

  • A outside 1 to 10divide/multiply A by 10 and adjust n.
  • E-07 on the calculatorwrite \(\times10^{-7}\).

Practice

Practice: easy, medium, hard

Easy

1
Write 8 600 000 000 in standard form.
Show answer

Answer \(8.6\times10^{9}\)

2
Write 0.017 in standard form.
Show answer

Answer \(1.7\times10^{-2}\)

3
Write \(9\times10^{3}\) as an ordinary number.
Show answer

Answer 9000

Medium

1
Write 15 060 in standard form.
Guide me

What is it asking? standard form

What do we know? keep the middle zero

Plan A = 1.506

Show working
  1. 4 jumps

Answer \(1.506\times10^{4}\)

2
Write \(450\times10^{-6}\) in standard form.
Guide me

What is it asking? standard form

What do we know? A = 450 is too big

Plan ÷ 100, power + 2

Show working
  1. \(4.5\times10^{-6+2}\)

Answer \(4.5\times10^{-4}\)

3
Write \(4.82\times10^{-3}\) as an ordinary number.
Guide me

What is it asking? an ordinary number

What do we know? power −3

Plan 3 places left

Show working
  1. 0.004 82

Answer 0.004 82

Hard

1
Write 28 795 701 in standard form, correct to 3 significant figures.
Guide me

What is it asking? standard form, 3 s.f.

What do we know? round first

Plan 28 800 000

Show working
  1. 28 800 000
  2. \(2.88\times10^7\)

Answer \(2.88\times10^{7}\)

PartAnswer / workingMarksGuidance
\(2.88\times10^{7}\)B1
Total: 1 mark
2
Earth's surface area is about 510 100 000 km² and water covers 70.8% of it. Area covered by water, in standard form to 3 s.f.?
Guide me

What is it asking? an area

What do we know? 70.8% of the total

Plan then round and write in standard form

Show working
  1. \(510\,100\,000\times0.708=361\,150\,800\)

Answer \(3.61\times10^{8}\) km²

PartAnswer / workingMarksGuidance
× 0.708M1
\(3.61\times10^{8}\)A1
Total: 2 marks
3
\(1.467\times10^{102}\) is written as an ordinary number. How many zeros follow the digit 7?
Guide me

What is it asking? a count

What do we know? the point moves 102 places

Plan 4, 6 and 7 use 3 of them

Show working
  1. 102 − 3

Answer 99

PartAnswer / workingMarksGuidance
99B1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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