Scientific notation writes a number as a single digit, a decimal part, and a power of ten. It exists to make very large and very small numbers writable, and it does a second job at the same time: it states exactly how many digits are meant to be trusted.
That second property is the reason scientists use it and the reason it is worth understanding beyond the writing convention.
What are the rules?
The leading part must be at least one and less than ten, and the exponent carries everything else. There is exactly one correct way to write any number, which is what makes two of them comparable at a glance.
| Written out | Scientific | Engineering |
|---|---|---|
| 47,000 | 4.7 × 10⁴ | 47 × 10³ |
| 0.00031 | 3.1 × 10⁻⁴ | 310 × 10⁻⁶ |
| 6,700,000 | 6.7 × 10⁶ | 6.7 × 10⁶ |
Engineering notation relaxes the first rule so that the exponent is always a multiple of three. That makes the leading part run from one up to a thousand, and it means every exponent corresponds to an SI prefix — kilo, mega, giga going up, milli, micro, nano coming down.
For anything that will be spoken or labelled, the engineering form is easier. A capacitor at 310 microfarads reads better than one at 3.1 times ten to the minus four farads.
Why is there an E in my spreadsheet?
Because plain text has no superscripts. The E form writes the exponent after a letter — so 3.1E-4 is the same number as the notation above — and it exists purely because keyboards and file formats cannot do raised digits.
It appears unbidden when a spreadsheet decides a number is too long for its column, which is how a long product code or an account number turns into something unrecognisable. The number has not been altered, but the displayed form has, and saving it back can make the loss real.
That is one more reason to store identifiers as text rather than as numbers, since nothing about an account number benefits from being treated arithmetically.
How does it show significant figures?
By writing only the digits that are meaningful. In ordinary notation 0.00450 relies on a convention to say the final zero is significant, and 4500 is genuinely ambiguous — it might be precise to two digits or to four.
Scientific notation removes the ambiguity because every digit written is a significant one. Four point five times ten cubed is two figures; four point five zero zero times ten cubed is four, and the two are different claims about the measurement.
The rules underneath are the familiar ones: leading zeros never count, digits between non-zero digits always do, and trailing zeros count only after a decimal point.
What happens in a computer?
The same idea in binary, with a fixed budget. A double-precision number holds a sign, an exponent and a fixed number of significant bits, which works out to roughly fifteen to seventeen significant decimal digits.
That budget is why very large integers lose their last digits and why some decimal fractions cannot be represented exactly. A tenth in binary is a recurring value in the same way a third is in decimal, so it is stored as an approximation.
The practical consequence is that comparing two computed decimals for exact equality is unreliable, and that money is better handled in whole units of the smallest denomination than as a fractional value.
When should you convert?
Whenever the number is being compared, published or measured. A column of figures in a consistent notation is readable; the same column mixing full numbers and exponent forms is not.
Going the other way matters just as often. A value arriving in exponent form from an export needs writing out before it goes into a document, and doing it by hand is where a factor of ten gets lost.
The rule that avoids most errors is to convert once, at the boundary, and keep one notation inside any single piece of work.
What are the prefixes worth knowing?
The ones that step by a thousand: kilo, mega, giga and tera going up, and milli, micro, nano and pico coming down. Each is three orders of magnitude from its neighbour, which is exactly why engineering notation aligns with them.
The two that break the pattern are centi and deci, which sit at two and one orders of magnitude and survive because centimetres are convenient. They are the reason the prefix system looks less regular than it is.
Questions people ask
Why do some fields prefer engineering notation? Because the exponent matches a unit prefix, so the number can be read aloud without translation.
How many digits does a spreadsheet keep? About fifteen significant ones, after which the trailing digits of a long number are lost rather than displayed differently.
Is 10⁰ equal to one? Yes, so a number between one and ten in scientific notation has an exponent of zero.
Which way does a negative exponent go? Smaller. A negative exponent means dividing by that power of ten.
Do I round before or after converting? After, and to the number of significant figures the measurement supports.
Why does my spreadsheet drop leading zeros? It read the value as a number. Formatting the column as text before pasting prevents it.
Convert rather than counting zeros. The scientific notation converter moves between the written-out form, the scientific form and the E notation a spreadsheet produces, in either direction.