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Cambridge IGCSE0607

Standard form

International Mathematics 0607 Chapter Notes

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Standard form
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In this chapter

  1. 1The form A × 10ⁿ
  2. 2Writing a number in standard form
  3. 3Writing a standard form number as an ordinary number
  4. 4Multiplying and dividing
  5. 5Adding and subtracting
  6. 6Using a graphic display calculator

1. The form A × 10ⁿ

A number in standard form is written A × 10ⁿ. The number A satisfies 1 leqslant A < 10, so it is at least 1 and less than 10. It has exactly one non-zero digit in front of the decimal point.

The power n is a positive or negative integer. A number of 10 or more has a positive power. A number between 0 and 1 has a negative power.

Both 45 × 10³ and 0.45 × 10⁵ equal 45 000, but neither is in standard form.

A × 10ⁿ, 1 leqslant A < 10, n an integer

Examiner insight

“Candidates must learn the correct way to express a number in standard form and to practise converting numbers to and from standard form.”

Worked example 11 mark

a × 10ⁿ represents a number written in standard form. Complete the inequality: … leqslant a < …

  1. 1

    1 leqslant a < 10 [1]

2. Writing a number in standard form

Place the decimal point after the first non-zero digit to make A. Count how many places the point moved. That count is the power.

For a number of 10 or more, the point moves left and the power is positive. For a number between 0 and 1, the point moves right and the power is negative.

Keep every significant figure. Do not round unless the question asks you to.

Examiner insight

“Those who knew, in principle, what standard form looks like, made errors by not having a single number in front of the decimal point or missing out the minus sign in the exponent.”

Worked example 12 marks

Write each number in standard form.

(a) 671 000 000

(b) 0.033

  1. 1

    (a) The point moves 8 places left: 6.71 × 10⁸ [1]

  2. 2

    (b) The point moves 2 places right: 3.3 × 10⁻² [1]

3. Writing a standard form number as an ordinary number

A positive power moves the decimal point to the right. A negative power moves it to the left. Fill the gaps with zeros.

Examiner insight

“Some candidates treated the standard form number as if the power was negative giving answers such as 0.00042.”

Worked example 11 mark

Write 6.4 × 10⁻³ as an ordinary number.

  1. 1

    The power is -3, so move the point 3 places left: 0.0064 [1]

4. Multiplying and dividing

Multiply or divide the A parts. Add the powers of 10 when multiplying. Subtract them when dividing.

The new A may fall outside the range 1 to 10. If it does, rewrite the answer in standard form as a last step.

(A × 10^(m)) × (B × 10ⁿ) = AB × 10^(m+n)

(A × 10^(m))/(B × 10ⁿ) = A/B × 10^(m-n)

Examiner insight

“Candidates must remember that the powers of 10 need to be added not multiplied.” “Some candidates changed the numbers into the same power, which is unnecessary when multiplying.”

Worked example 12 marks

P = 3.2 × 10²² and Q = 4 × 10²¹.

Calculate PQ. Give your answer in standard form.

  1. 1

    PQ = (3.2 × 4) × 10²²⁺²¹ = 12.8 × 10⁴³ [1]

  2. 2

    12.8 × 10⁴³ = 1.28 × 10⁴⁴ [1]

5. Adding and subtracting

Write both numbers with the same power of 10. Choose the larger power. Then add or subtract the A parts and keep the power.

Check that the answer is in standard form. If not, rewrite it.

Examiner insight

“Candidates needed to write both numbers to the same power and then ensure that their final answer was in standard form.”

Worked example 12 marks

Work out 4.2 × 10¹⁶ + 3.1 × 10¹⁵. Give your answer in standard form.

  1. 1

    3.1 × 10¹⁵ = 0.31 × 10¹⁶ [1]

  2. 2

    4.2 × 10¹⁶ + 0.31 × 10¹⁶ = 4.51 × 10¹⁶ [1]

6. Using a graphic display calculator

On papers 3 and 4 a graphic display calculator is allowed. Enter each number with the calculator's power of 10 key, marked ×10ˣ or EE: for example 2.5, then the key, then −12. Use brackets round each standard form number.

The display may show an abbreviation such as 1.74E−12. Rewrite it in full as 1.74 × 10⁻¹².

Core candidates calculate with standard form only on papers where calculators are allowed.

Examiner insight

“Standard form should be correctly written as 6 × 10–3 rather than the abbreviated calculator notation.” “Some subtracted 7.6 from 2.5 and then subtracted the powers of 10.”

Worked example 11 mark

Work out (2.5 × 10⁻¹²) - (7.6 × 10⁻¹³).

  1. 1

    Enter (2.5 × 10⁻¹²) - (7.6 × 10⁻¹³) and read off 1.74E-12, which is 1.74 × 10⁻¹² [1]

Recap

  • Standard form is A × 10ⁿ with 1 leqslant A < 10 and n an integer.
  • Numbers of 10 or more have a positive power. Numbers between 0 and 1 have a negative power.
  • Into standard form: count the places the decimal point moves. Out of it: move the point the other way.
  • Multiply: multiply the A parts and add the powers. Divide: divide the A parts and subtract the powers.
  • Add or subtract: use the same power first.
  • After every calculation, check A is still between 1 and 10.
  • Core candidates calculate with standard form only on calculator papers. Write calculator displays out in full.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Which of these is in standard form? 0.45 × 10⁵, 45 × 10³, 4.5 × 10⁴.1 mark
  2. Do not use a calculator. Write 7 300 000 in standard form.1 mark
  3. Do not use a calculator. Write 0.000 52 in standard form.1 mark
  4. Write 3.8 × 10⁻³ as an ordinary number.1 mark
  5. Write 36 × 10⁻⁵ in standard form.2 marks
  6. Do not use a calculator. Work out (4 × 10⁵) × (3 × 10⁴). Give your answer in standard form.2 marks
  7. Do not use a calculator. Work out (6 × 10³)/(4 × 10⁻⁵). Give your answer in standard form.2 marks
  8. Do not use a calculator. Work out 5.2 × 10⁶ + 8 × 10⁵. Give your answer in standard form.2 marks
  9. Do not use a calculator. A small memory chip stores 4 × 10^(y) bytes and a large chip stores 8 × 10^(y+1) bytes, where y is a positive integer. Give each answer in standard form, in terms of y. (a) Find the total number of bytes stored by one small chip and one large chip. (b) A server holds 5 × 10³ large chips. Find the number of bytes they store altogether.4 marks
  10. A thin film is made of identical layers, each 3.4 × 10⁻¹⁰ m thick. The film is 1.7 × 10⁻³ m thick. Use a calculator. (a) Find the number of layers. Give your answer in standard form. (b) Each layer has mass 7.6 × 10⁻⁷ g. Find the mass of the film in g. Give your answer as an ordinary number. (c) Another 6.5 × 10⁵ layers are added. Find the new thickness of the film in m. Give your answer in standard form.5 marks

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