Concept · Basic Math
What Is Scientific Notation? A Plain Explanation
Scientific notation is a way of writing numbers by their size and their digits separately. The digits go in front; a power of 10 says how far the decimal point has to travel.
The problem it solves
Some numbers are awkward to write. The mass of the Sun is about 1,989,000,000,000,000,000,000,000,000,000,000 kilograms, and a hydrogen atom is about 0.000000000000000000000000167 kilograms. Counting the zeros in either is slow and easy to get wrong, and the zeros hide the one thing you actually want to know, which is how big the number is.
So what is scientific notation? It is a way of splitting a number into two parts: the digits that matter, written as a number from 1 up to (but not including) 10, and a power of 10 that records the size. The Sun's mass becomes 1.989 × 10³⁰ kg. The hydrogen atom becomes 1.67 × 10⁻²⁷ kg. Both are short, and the exponent shows the scale at a glance.
The shape: a × 10ⁿ
Every number in scientific notation has the same form, a × 10ⁿ.
| Part | Name | Rule |
|---|---|---|
| a | the coefficient (or mantissa) | at least 1 and less than 10 |
| 10 | the base | always 10 |
| n | the exponent | any whole number, positive or negative |
The rule on the coefficient is what makes the notation consistent. 45 × 10³ is a correct value, but it is not scientific notation, because 45 is not below 10. Written properly it is 4.5 × 10⁴, and there is only one proper way to write any given number.
What the exponent counts
The exponent is the number of places the decimal point moves. Multiplying by 10 shifts the point one place right, and multiplying by 10ⁿ shifts it n places right. So a positive exponent means the real number is large, and a negative exponent means it is small.
- 3.2 × 10⁴ = 32000positive exponent: the point moves 4 places right, a big number
- 3.2 × 10⁰ = 3.2zero exponent: the point does not move
- 3.2 × 10⁻⁴ = 0.00032negative exponent: the point moves 4 places left, a small number
Notice that the sign says which side of 1 the number sits on, and the size of the exponent says how far. The exponent is not a count of zeros: 45000 has three zeros but its exponent is 4, because the point travels four places.
Why scientists use it
Beyond saving space, the notation makes comparison easy. Two numbers in scientific notation can be compared by their exponents first. 6 × 10⁸ is bigger than 9 × 10⁷ because 8 is greater than 7, even though 9 is greater than 6. This is called comparing the order of magnitude, and it is the quickest way to see how far apart two quantities are.
It also separates precision from scale. The digits in the coefficient are the significant figures of a measurement, and the exponent is only the scale. Writing 1.50 × 10³ rather than 1500 tells the reader that the trailing zero was measured, which a plain 1500 cannot say.
Scientific notation versus E notation
Calculators and spreadsheets often show 3.2E5 or 3.2e5. The letter is just shorthand for × 10 to the power, so 3.2e5 means 3.2 × 10⁵ and 320000. It is the same idea typed on one line, which is why this site's calculator reads either form.
A quick sanity check
To check a conversion, ask whether the exponent's sign matches the number's size. If the original is 10 or more, the exponent must be positive. If it is between 0 and 1, the exponent must be negative. A mismatch means the decimal point was moved the wrong way, the most common slip with scientific notation.
Questions about what is scientific notation
What is scientific notation in simple terms?
A way of writing a number as a digit string between 1 and 10 multiplied by a power of 10. 45000 is 4.5 × 10⁴. It keeps very large and very small numbers short.
Why must the first number be between 1 and 10?
So that every number has exactly one correct form. Without the rule, 45 × 10³, 4.5 × 10⁴ and 0.45 × 10⁵ would all be acceptable, and numbers would be harder to compare.
What does a negative exponent mean?
The number is smaller than 1. The decimal point moves left by the size of the exponent: 3.2 × 10⁻⁴ is 0.00032.
Is 3.2e5 the same as 3.2 × 10⁵?
Yes. The e stands for ×10 to the power of, so 3.2e5 is 320000. It is the form calculators and spreadsheets display.