Scientific notation: a × 10ⁿ for very large and very small numbers

Count the places the point moves

Your book opens Unit 2 by saying logarithms make calculations with very large or very small numbers easier. The first tool is scientific notation.

A number in scientific notation is a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. The number a is called the coefficient.

Move the decimal point until one non-zero digit is left of it, and count the places. Moving left gives a positive n; moving right gives a negative n.

Going back, a positive exponent moves the point right and a negative one moves it left. So 3.47 × 10⁶ = 3,470,000.

Number bigger than 1: n is positive. Number smaller than 1: n is negative. The coefficient always sits between 1 and 10.

Key terms

Scientific notation

Writing huge or tiny numbers briefly.

Coefficient

The digits part of the number.

Exponent

How many places the point moved.

Ordinary notation

The usual way of writing a number.

Short questions with model answers

1. Q1. Example 1: convert 78,000,000 to scientific notation.

   • 7.8
   • 7 places to the left

   7.8 × 10⁷.

2. Q2. Example 4: convert 6.23 × 10⁻⁴ to ordinary notation.

   • coefficient 6.23, exponent −4
   • 6.23 → 0.000623

   0.000623.

Common mistakes

• ✗ 0.0000000315 = 3.15 × 10⁸.

  ✓ The number is less than 1, so n is negative: 10⁻⁸.

• ✗ Writing 78,000,000 as 78 × 10⁶.

  ✓ The coefficient must satisfy 1 ≤ a < 10: 7.8 × 10⁷.

• ✗ Counting only the zeros: 0.0000000315 has 7 zeros after the point, so 10⁻⁷.

  ✓ Count the moves to reach 3.15: the point passes the 3 too, so 8 places.

MCQs

1. 1. The standard form of 5.2 × 10⁶ is:

   1. (a) 520,000
   2. (b) 5,200,000
   3. (c) 52,000,000
   4. (d) 52,0000
   Show answer

   (b) Move the point 6 places right (Review Exercise 2 i).

2. 2. Scientific notation of 0.00034 is:

   1. (a) 3.4 × 10⁴
   2. (b) 3.4 × 10⁻³
   3. (c) 34 × 10⁻⁵
   4. (d) 3.4 × 10⁻⁴
   Show answer

   (d) 4 places right gives 10⁻⁴ (Review Exercise 2 ii).

3. 3. In 3.15 × 10⁻⁸, the coefficient is:

   1. (a) 3.15
   2. (b) 10
   3. (c) −8
   4. (d) 315
   Show answer

   (a) a = 3.15, and 1 ≤ 3.15 < 10 (p. 21).

4. 4. Which is in correct scientific notation?

   1. (a) 73 × 10³
   2. (b) 0.65 × 10²
   3. (c) 6.779 × 10³
   4. (d) 40.075 × 10⁶
   Show answer

   (c) Only 6.779 has 1 ≤ a < 10.

5. 5. 3.47 × 10⁶ in ordinary notation is:

   1. (a) 347,000
   2. (b) 34,700,000
   3. (c) 0.00000347
   4. (d) 3,470,000
   Show answer

   (d) 6 places right (Example 3).

Quick revision

• Scientific notation is a × 10ⁿ with 1 ≤ a < 10 and n an integer.
• Numbers above 1 get a positive n; numbers below 1 a negative n.
• A positive exponent moves the point right; a negative one moves it left.