What this calculator does
Standard form writes any number as a coefficient between 1 and 10 times a power of ten. Multiplying and dividing are easy in this form, since the exponents simply add or subtract, but adding requires matching the exponents first.
The difference shows in the arithmetic. Multiplying 5 × 10³ by 2 × 10² gives 1.000 × 10⁶ immediately, while adding them requires converting to the same power first and gives 5.200 × 10³.
The formula
Both numbers are expanded, the operation is applied, and the result is renormalised so the coefficient falls between 1 and 10.
| Term | Meaning |
|---|---|
| Standard form | A coefficient between 1 and 10 multiplied by a power of ten. |
| Renormalising | Adjusting the coefficient and exponent so the coefficient is in range. |
| E-notation | The same thing written as 5.2e3, which calculators and computers use. |
The inputs explained
| Field | What to enter |
|---|---|
| Coefficient a | Coefficient of the first number. |
| Exponent m | Exponent of the first number. |
| Operation | Which operation to perform. |
| Coefficient b | Coefficient of the second number. |
| Exponent n | Exponent of the second number. |
When to use it
Science calculations
Physical constants and measurements are almost always written in standard form.
Checking a calculator result
E-notation output is easy to misread by an order of magnitude.
Learning the rules
Multiplication and addition behave quite differently in this form.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does each operation behave?
The same pair under all four operations.
| Operation | Result in standard form | Result, unrounded |
|---|---|---|
| Add | 5.200 × 10^3 | 5,200 |
| Subtract | 4.800 × 10^3 | 4,800 |
| Multiply | 1.000 × 10^6 | 1,000,000 |
| Divide | 2.500 × 10^1 | 25 |
Questions
Why is multiplication easier than addition?
Because exponents add under multiplication, so the two parts can be handled separately. Addition requires both numbers to share an exponent before the coefficients can be combined, which means converting one of them first.
Why must the coefficient be between 1 and 10?
So that every number has exactly one standard form. Without the constraint, 5,200 could be written as 5.2 × 10³, 52 × 10² or 0.52 × 10⁴, and comparisons would be needlessly awkward.
Is scientific notation the same thing?
Yes. Standard form is the British term and scientific notation the American one for the identical convention. Engineering notation differs slightly, restricting exponents to multiples of three.
What does e mean on a calculator?
It is shorthand for "times ten to the power of". A display of 5.2e3 means 5.2 × 10³, or 5,200. It has nothing to do with the mathematical constant e.
For general percentage work, see the percentage calculator. For significant figures in measurement, see the percent error calculator.