Scientific Notation (Standard Form)

What is scientific notation?

Some numbers are very big. Some are very small. Writing all their zeros is slow, and it is easy to make a mistake.

Scientific notation (also called standard form) writes a number as:

a × 10ⁿ

• a is a number from 1 up to (but not including) 10. So 1 ≤ a < 10.
• n is an integer (a whole number, which can be positive, zero or negative). It is called the power or exponent.

Examples: 5000 = 5 × 10³. 0.07 = 7 × 10⁻². 6.2 = 6.2 × 10⁰.

Not in scientific notation: 25 × 10³ (25 is bigger than 10) and 0.5 × 10⁴ (0.5 is less than 1).

Powers of ten

A power of ten is 10 multiplied by itself a number of times.

• 10¹ = 10, 10² = 100, 10³ = 1000, 10⁶ = 1 000 000 (one million).
• 10⁰ = 1.
• 10⁻¹ = 1/10 = 0.1, 10⁻² = 1/100 = 0.01, 10⁻³ = 0.001.

Easy rule: for a positive power, n is the number of zeros after the 1. For a negative power, n tells how many places the 1 sits after the decimal point.

The same rules of exponents work here: 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ and 10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ.

Converting: normal form to scientific notation and back

Big numbers (positive power)

1. Put the decimal point after the first non-zero digit.
2. Count how many places it moved to the left. That is n.
3. Drop the extra zeros at the end.

4 560 000 → 4.56 (moved 6 places left) → 4.56 × 10⁶.

Small numbers (negative power)

1. Move the point to the right until it is after the first non-zero digit.
2. Count the places. n is that count with a minus sign.

0.000 302 → 3.02 (moved 4 places right) → 3.02 × 10⁻⁴.

Back to normal form

Positive n: move the point n places right. 7.1 × 10⁵ = 710 000. Negative n: move it left. 2.5 × 10⁻³ = 0.0025.

Operations in scientific notation

Multiply

Multiply the a's. Add the powers. (3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹.

Divide

Divide the a's. Subtract the powers. (8 × 10⁹) ÷ (2 × 10³) = 4 × 10⁶.

Fix the front number

If the answer's front number is not between 1 and 10, adjust it. (5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2 × 10⁶ (20 = 2 × 10¹, so add 1 to the power).

Add or subtract

First make the powers the same. 3 × 10⁵ + 4 × 10⁴ = 3 × 10⁵ + 0.4 × 10⁵ = 3.4 × 10⁵.

On a calculator, the key EXP or ×10ˣ enters the power. 3E8 on a screen means 3 × 10⁸.

Estimating with powers of 10 and unit prefixes

To estimate, round a number to one digit times a power of ten. Then compare. Example: a city has about 2 × 10⁷ people and a village about 4 × 10³. The city has about (2 × 10⁷) ÷ (4 × 10³) = 0.5 × 10⁴ = 5000 times more people.

The order of magnitude is the power of ten closest to a number. 3 × 10⁸ is of order 10⁸.

SI unit prefixes are names for powers of ten:

• giga (G) = 10⁹, mega (M) = 10⁶, kilo (k) = 10³
• centi (c) = 10⁻², milli (m) = 10⁻³, micro (µ) = 10⁻⁶, nano (n) = 10⁻⁹

So 5 km = 5 × 10³ m, 3 mg = 3 × 10⁻³ g and 20 nm = 2 × 10⁻⁸ m.

Try it: write the world in powers of ten

In the 3D free play, set the front number and the power. Watch the orange marker move up the size ladder from an atom (about 10⁻¹⁰ m) to the Sun (about 10⁹ m). Before you move the slider, predict which object you will land on. Then check.

Key formulas and definitions

• Scientific notation: a × 10ⁿ, with 1 ≤ a < 10 and n an integer
• Big number → point moves left → n positive
• Small number → point moves right → n negative
• Multiply: (a × 10ᵐ)(b × 10ⁿ) = ab × 10ᵐ⁺ⁿ
• Divide: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ
• Prefixes: G 10⁹, M 10⁶, k 10³, c 10⁻², m 10⁻³, µ 10⁻⁶, n 10⁻⁹

Worked examples

Common mistakes

• Leaving the front number outside 1 to 10, like 45 × 10³. Fix it: 4.5 × 10⁴.
• Giving a small number a positive power. 0.003 is 3 × 10⁻³, not 3 × 10³. Small numbers have negative powers.
• Multiplying the powers instead of adding them. 10³ × 10⁵ = 10⁸, not 10¹⁵.
• Adding numbers with different powers directly. 2 × 10³ + 3 × 10⁴ is not 5 × 10⁷. Make the powers equal first: 0.2 × 10⁴ + 3 × 10⁴ = 3.2 × 10⁴.