How to Write Numbers in Scientific Notation, Step by Step

The format

a × 10ⁿ, with 1 ≤ |a| < 10 and n a whole numbera is the coefficient (or mantissa); n is the exponent.

The coefficient must have exactly one non-zero digit before the decimal point. So 4.56 × 10⁷ is correct, but 45.6 × 10⁶ and 0.456 × 10⁸ are not in proper scientific form, even though they equal the same number.

A positive exponent means the number is 10 or more. A negative exponent means it is smaller than 1. An exponent of 0 means the number is already between 1 and 10, because 10⁰ = 1.

Converting to and from scientific notation

1. Move the decimal point until exactly one non-zero digit sits to its left.
2. Count how many places you moved it. That count is the size of the exponent.
3. If you moved the point left (the original number was large), the exponent is positive. If you moved it right (the number was small), it is negative.

Worked example, large: 45,600,000. The decimal point sits after the final zero. Moving it 7 places left gives 4.56, so the number is 4.56 × 10⁷.

Worked example, small: 0.00032. Moving the point 4 places right gives 3.2, so the number is 3.2 × 10⁻⁴.

To convert back, move the decimal point the number of places the exponent shows, in the direction of its sign, and pad with zeros. The speed of light in a vacuum is exactly 299,792,458 metres per second, which is 2.99792458 × 10⁸ m/s, or about 3.00 × 10⁸ m/s when rounded.

Multiplying and dividing

Multiply or divide the coefficients, then add or subtract the exponents. This follows from the laws of exponents: 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ and 10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ.

Multiplying: (3 × 10⁴) × (2.5 × 10⁶). Coefficients: 3 × 2.5 = 7.5. Exponents: 4 + 6 = 10. The product is 7.5 × 10¹⁰, which is 75,000,000,000.

Dividing: (6 × 10⁸) ÷ (1.5 × 10³). Coefficients: 6 ÷ 1.5 = 4. Exponents: 8 − 3 = 5. The quotient is 4 × 10⁵, or 400,000.

Renormalising: (4 × 10³) × (5 × 10²) gives 20 × 10⁵, but 20 is not between 1 and 10. Move one place and adjust: 20 × 10⁵ = 2 × 10⁶.

Adding and subtracting

You cannot add the coefficients directly unless the exponents match. First rewrite one number so both use the same power of ten, then combine the coefficients and keep the exponent.

Worked example: 3.2 × 10⁵ + 4.1 × 10⁴. Rewrite 3.2 × 10⁵ as 32 × 10⁴. Then 32 + 4.1 = 36.1, so the sum is 36.1 × 10⁴. Renormalise to get 3.61 × 10⁵. In ordinary numbers, 320,000 + 41,000 = 361,000.

A common error is to add the exponents when adding numbers (or to multiply coefficients when adding). Exponents are added only when you multiply.

E notation and significant figures

Calculators, spreadsheets and programming languages usually write scientific notation as an E. The display 4.56E7 means 4.56 × 10⁷, and 3.2E-4 means 3.2 × 10⁻⁴. The E stands for "exponent" and is not the mathematical constant e, which is about 2.718.

Scientific notation is also a clean way to show how precise a number is. Written as 4.50 × 10³, the number has three significant figures, because the trailing zero is deliberate. Plain 4500 leaves it unclear whether the zeros are measured or just placeholders.

• Keep as many digits in the coefficient as you have significant figures.
• Round only at the end of a multi-step calculation.
• Check the answer's size: a large positive exponent should match a very large number.

To convert or check any value, use the scientific notation calculator. If your course uses the UK term, the standard form calculator does the same job.