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Mathematics 4024

Quick revision notes · Mathematics 4024 · Chapter 3 of 35

Standard Form

Writing very large and very small numbers as A × 10n, ordering them, and calculating with them.

4 sectionsWorked examples Common mistakes
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At a glance

Writing very large and very small numbers as A × 10n, ordering them, and calculating with them.

  1. What standard form is
  2. Ordering numbers in standard form
  3. Calculating without a calculator
  4. Using a calculator and real problems

1What standard form is

  • A number in standard form is A × 10n with 1 ⩽ A < 10 and n a whole number.
  • Large numbers have a positive power: 4 560 000 = 4.56 × 106.
  • Numbers less than 1 have a negative power: 0.00072 = 7.2 × 10−4.
  • The power is how many places the decimal point moves to get A.

How to do it

  1. Write the digits with the point after the first non-zero digit (that is A).
  2. Count the places the point moved.
  3. Moved left (big number) → positive power; moved right (small number) → negative power.
OrdinaryStandard form
300 0003 × 105
45 6004.56 × 104
0.055 × 10−2
0.000 3073.07 × 10−4

Worked example

Write 0.000 307 in standard form.

  1. First non-zero digit is 3: A = 3.07.
  2. The point moves 4 places to the right.

3.07 × 10−4

2Ordering numbers in standard form

  • Compare the powers first: the bigger power is the bigger number (for positive numbers).
  • If the powers are equal, compare the A values.
  • Negative powers: 10−2 is bigger than 10−3.

Worked example

Order, smallest first: 3.1 × 10−2, 9.8 × 10−3, 1.2 × 10−2.

  1. 10−3 is the smallest power, so 9.8 × 10−3 is smallest.
  2. Then 1.2 × 10−2 before 3.1 × 10−2.

9.8 × 10−3, 1.2 × 10−2, 3.1 × 10−2

3Calculating without a calculator

  • Multiply: multiply the A values and add the powers.
  • Divide: divide the A values and subtract the powers.
  • Then fix the answer if A is not between 1 and 10: 12 × 109 = 1.2 × 1010; 0.5 × 103 = 5 × 102.
  • Add or subtract: change both numbers to ordinary numbers (or to the same power), then add.

Formulas

(a × 10m)(b × 10n) = ab × 10m+n
(a × 10m) ÷ (b × 10n) = (a/b) × 10m−n

Worked example

Work out (6 × 105) ÷ (1.5 × 10−2).

  1. 6 ÷ 1.5 = 4.
  2. 105 − (−2) = 107.

4 × 107

Exam tip: Adding: 5 × 103 + 7 × 102 = 5000 + 700 = 5700 = 5.7 × 103.

4Using a calculator and real problems

  • Use the ×10x (EXP) key: type 3.2 ×10x 5, not 3.2 × 10 ^ 5 (which can go wrong with division).
  • Give the final answer in standard form if the question uses it, rounded to 3 significant figures.
  • Typical contexts: distances in space, masses of atoms, populations, speed of light (3 × 108 m/s).

Worked example

Light travels 3 × 108 m each second. How long does it take to travel 1.5 × 1011 m?

  1. time = distance ÷ speed.
  2. 1.5 ÷ 3 = 0.5 and 1011−8 = 103.
  3. 0.5 × 103 = 500.

500 seconds

✗Common mistakes

Don’t writeWrite instead
Leaving 45.6 × 105 as the answer A must be between 1 and 10: 4.56 × 106
0.00072 = 7.2 × 104 Small numbers need a negative power: 7.2 × 10−4
Adding the powers when adding numbers Only multiplication adds the powers
Calculator answer 2.4E7 copied as 2.47 It means 2.4 × 107

★In the exam

  • Read whether the answer must be in standard form — then give it in standard form.
  • On the non-calculator paper, show the separate steps for the numbers and the powers.
  • Check size: a speed or distance should be sensible after you divide.