Scientific notation: the Sun's distance without a row of zeros

The problem

The average distance from the Sun to the Earth is 138,000,000 km. A hydrogen atom is about 0.000 000 000 052 m across. Try comparing them, or multiplying with them. The zeros get in the way.

Scientific notation writes a number as one non-zero digit before the decimal point, times a whole-number power of 10.

Move the decimal point until one non-zero digit is left of it, and count the places. Moving left gives a positive power: 1.38 × 10⁸ km. Moving right gives a negative power: 5.2 × 10⁻¹¹ m.

To add or subtract, first make the powers of ten the same. To multiply, multiply the numbers and add the powers; to divide, divide the numbers and subtract the powers.

Scientific notation is one non-zero digit before the decimal, times a power of ten. Powers must match before you add or subtract; for × and ÷ you add or subtract the powers.

Key terms

Scientific notation

A number from 1 to 9.99… times a whole-number power of 10.

Numerical part

The number in front of the power of 10, with one non-zero digit before the decimal.

Positive exponent

The places the decimal moved to the left.

Negative exponent

The places the decimal moved to the right; the value is less than one.

Short questions with model answers

1. Q1. Write the Sun–Earth distance, 138,000,000 km, in scientific notation.

   • 138,000,000 → 1.38
   • = 1.38 × 10⁸ km

   1.38 × 10⁸ km

2. Q2. Example 1.1 (b): 2.57 × 10⁻² mm − 3.43 × 10⁻³ mm. Give the answer in metres.

   • 3.43 × 10⁻³ mm = 0.343 × 10⁻² mm
   • (2.57 − 0.343) × 10⁻² mm = 2.227 × 10⁻² mm
   • 2.227 × 10⁻² × 10⁻³ m = 2.227 × 10⁻⁵ m

   2.227 × 10⁻⁵ m

3. Q3. Example 1.2 (a): (4 × 10³ kg)(6 × 10⁶ m).

   • (4 × 6) × 10³⁺⁶ kg m = 24 × 10⁹ kg m
   • = 2.4 × 10¹⁰ kg m

   2.4 × 10¹⁰ kg m

Common mistakes

• ✗ Adding 5.123 × 10⁴ and 3.28 × 10⁵ as 8.403 × 10⁴ or 8.403 × 10⁹.

  ✓ Make the powers equal first; only then add the numbers. Powers are added only when multiplying.

• ✗ Writing 0.00534 m as 5.34 × 10³ m.

  ✓ A number smaller than one needs a negative power: 5.34 × 10⁻³ m.

• ✗ Writing units as '5 kgs', '60 sec' or '10 gm'.

  ✓ Use symbols with no plural: 5 kg, 60 s, 10 g.

MCQs

1. 1. Quick Quiz (p. 11): 0.00534 m in scientific notation is:

   1. (a) 5.34 × 10³ m
   2. (b) 53.4 × 10⁻⁴ m
   3. (c) 0.534 × 10⁻² m
   4. (d) 5.34 × 10⁻³ m
   Show answer

   (d) Move the decimal 3 places right to leave 5 before it, so the power is −3.

2. 2. Quick Quiz (p. 11): 186000 s in scientific notation is:

   1. (a) 1.86 × 10⁵ s
   2. (b) 18.6 × 10⁴ s
   3. (c) 1.86 × 10⁻⁵ s
   4. (d) 186 × 10³ s
   Show answer

   (a) The decimal moves 5 places left; only 1.86 has one digit before it.

3. 3. 7 μμs should be written as:

   1. (a) 7 ns
   2. (b) 7 ps
   3. (c) 7 ms
   4. (d) 7 fs
   Show answer

   (b) 10⁻⁶ × 10⁻⁶ = 10⁻¹², which is pico; compound prefixes are not allowed.

4. 4. Numerical 1.4 (a): (5 × 10⁴ m) × (3 × 10⁻² m) =

   1. (a) 15 × 10⁻⁸ m²
   2. (b) 1.5 × 10² m²
   3. (c) 1.5 × 10³ m²
   4. (d) 8 × 10² m²
   Show answer

   (c) 5 × 3 = 15 and 4 + (−2) = 2, so 15 × 10² = 1.5 × 10³ m².

5. 5. Which is written correctly in SI?

   1. (a) 5 kgs
   2. (b) 60 sec
   3. (c) 10 gm
   4. (d) 100 N
   Show answer

   (d) N is the symbol for newton, with no plural; the others use abbreviations or plurals.

Quick revision

• One non-zero digit before the decimal × a power of ten: 1.38 × 10⁸ km, 5.2 × 10⁻¹¹ m.
• Match the powers before + and −; add or subtract the powers for × and ÷.
• Write SI with symbols, no plurals and no double prefixes.