Why scientific notation exists
Write out the mass of an electron in full and you get 0.000000000000000000000000000000910938 kilograms. Nobody can read that, count the zeros reliably, or tell at a glance how it compares with something else. Scientific notation solves all three problems by splitting every number into two parts: the digits that carry the information, and a power of ten that says where the decimal point belongs. The electron mass becomes 9.10938 × 10−31 kg.
The convention is that the coefficient has exactly one non-zero digit before the decimal point, so it is at least 1 and less than 10. That constraint makes the representation unique: 93,000,000 is 9.3 × 107 and nothing else. Without it, 93 × 106 and 0.93 × 108 would be equally valid, and comparing two numbers would mean normalising them first anyway.
There is a second benefit that matters as much in laboratory work: scientific notation makes significant figures unambiguous. Written as 93,000,000, you cannot tell whether the measurement is good to two digits or to eight. Written as 9.3 × 107, it declares two significant figures; 9.300 × 107 declares four. Trailing zeros in a plain decimal are ambiguous, and in scientific notation they are not.
Scientific, E and engineering notation
Scientific notation is m × 10e with 1 ≤ |m| < 10. To convert, move the decimal point until one non-zero digit remains in front of it, then count the moves: left moves give a positive exponent, right moves a negative one. 93,000,000 needs seven left moves, so the exponent is 7. 0.000045 needs five right moves, so the exponent is −5.
E notation writes the same thing in plain text where superscripts are unavailable: 9.3E+7 and 4.5E−5. This is what spreadsheets display, what most programming languages accept as input, and what appears in CSV exports. It means exactly the same thing; the E stands for exponent, not for the mathematical constant e, which is a genuine source of confusion since 1E3 is 1,000 and not 20.09.
Engineering notation restricts the exponent to multiples of three, which lets the coefficient run from 1 up to just under 1,000. 0.000045 becomes 45 × 10−6 rather than 4.5 × 10−5. The reason is that every SI prefix corresponds to a power of ten that is a multiple of three, so an engineering exponent maps directly onto a unit name: 10−6 is micro, so 45 × 10−6 farads is 45 microfarads, which is what is printed on the component. Electronics, mechanical and civil work all default to this form for that reason.
Arithmetic is where the notation earns its keep. To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents. Both follow directly from the laws of exponents and neither requires you to write a single zero. Addition and subtraction are the awkward pair, because the powers of ten must match first — you cannot add 3 × 108 to 2 × 105 until one is rewritten with the other's exponent.
Worked example: converting, multiplying and adding
Convert 93,000,000. Put the decimal point after the 9: 9.3. Count how far it moved to get there — from after the final zero to after the 9 is seven places left. So 93,000,000 = 9.3 × 107. In E notation, 9.3E+7. In engineering notation, the nearest multiple of three at or below 7 is 6, so the coefficient becomes 93 and the answer is 93 × 106 — 93 million, or 93 mega-anything.
Convert 0.000045. Move the point right until one non-zero digit sits in front: 4.5, after five moves. Right moves are negative, so 4.5 × 10−5. Engineering form takes the nearest multiple of three at or below −5, which is −6, giving 45 × 10−6.
Multiply (3 × 108) × (2 × 10−3). Coefficients: 3 × 2 = 6. Exponents: 8 + (−3) = 5. So the answer is 6 × 105 = 600,000. Note that no normalisation was needed because 6 is already between 1 and 10. Had the coefficients multiplied to 60, you would shift one place: 60 × 105 = 6 × 106.
Add 3 × 108 and 2 × 105. The exponents differ, so rewrite the smaller one to match the larger: 2 × 105 = 0.002 × 108. Now the coefficients can be added: 3 + 0.002 = 3.002, so the answer is 3.002 × 108. The rule for the rewrite is that raising the exponent by k moves the decimal point k places left in the coefficient, which keeps the product unchanged.
A precision warning falls straight out of that. Add 3 × 108 and 2 × 10−12 and the second value would sit twenty places down the coefficient. Standard double-precision arithmetic carries about sixteen significant digits, so the smaller number vanishes entirely and the computed sum is exactly 3 × 108. That is not a bug in the calculator; it is the reason numerical software adds long lists of values smallest-first.
Reading the exponent
The exponent is the order of magnitude, and it is the first thing to check. A positive exponent means a number bigger than 1, a negative exponent means a number between 0 and 1, and an exponent of zero means the number itself is between 1 and 10. If you converted 0.000045 and got a positive exponent, the decimal point moved the wrong way.
The exponent is also one less than the digit count for whole numbers: 9.3 × 107 has eight digits before the decimal point. That relationship is exactly the base-10 logarithm, and the logarithm calculator gives it with the fractional part included, which is how you find the digit count of numbers far too large to write out.
Comparing two numbers becomes a two-stage check: compare exponents first, and only compare coefficients if the exponents are equal. 8.9 × 105 is smaller than 1.2 × 106 despite the larger coefficient. This is the main practical reason for insisting on a normalised coefficient.
Watch the coefficient's digit count, because it declares your precision. Reporting 9.30000 × 107 for a measurement made to two significant figures is a false claim about the instrument. Multiplication and division carry the smaller number of significant figures through to the answer, so 3.0 × 108 times 2.15 × 10−3 should be reported as 6.5 × 105, not 6.45 × 105. The significant figures calculator applies those rules properly.
Powers of ten and their SI prefixes
| Power of ten | Plain value | Prefix | Symbol |
|---|---|---|---|
| 1012 | 1 000 000 000 000 | tera | T |
| 109 | 1 000 000 000 | giga | G |
| 106 | 1 000 000 | mega | M |
| 103 | 1 000 | kilo | k |
| 100 | 1 | — | — |
| 10−3 | 0.001 | milli | m |
| 10−6 | 0.000 001 | micro | µ |
| 10−9 | 0.000 000 001 | nano | n |
| 10−12 | 0.000 000 000 001 | pico | p |
Case matters: M is mega and m is milli, a factor of a billion apart. The kilo symbol is a lowercase k in the SI, even though a capital K is common in informal writing.
Mistakes that change the answer by a factor of ten
- Counting the moves in the wrong direction. Moving the point left gives a positive exponent; moving right gives a negative one. Check the sign against whether the number is bigger or smaller than 1 before anything else.
- Leaving the coefficient outside 1 to 10. 0.93 × 108 and 93 × 106 are correct values but not scientific notation. Only the engineering form deliberately relaxes this.
- Adding without matching the exponents. 3 × 108 + 2 × 105 is not 5 × 1013. Exponents add for multiplication only.
- Reading E as the constant e. 2E3 is 2,000, not 2 × 2.7183. E notation is purely a typographic device for powers of ten.
- Inventing significant figures. Converting a two-figure measurement into a coefficient with six digits claims precision the measurement never had. Round the coefficient to the figures you actually know.
- Assuming a spreadsheet's display is the stored value. A cell showing 9.3E+07 may hold 93,000,123. Display formatting rounds; it does not change what is stored, and later arithmetic uses the stored value.
Related tools
Scientific notation is powers of ten, so the exponent calculator is the underlying operation: 107 is what the notation is multiplying by, and the same rules for negative and fractional exponents apply. Going the other way, the base-10 logarithm of a number is its exponent plus the log of its coefficient, which is how you recover an order of magnitude from a value too large to display.
Deciding how many digits to keep in the coefficient is a separate discipline from the conversion itself. The significant figures calculator applies the multiplication, division, addition and rounding rules that determine how many digits a computed result is entitled to.
Computing uses the same idea in base 2 rather than base 10 — a floating-point number is a binary mantissa times a power of two — and the number base conversion calculator moves between the bases where that matters. For very small or very large values expressed as fractions rather than decimals, the fraction to decimal calculator gives the decimal form to convert from, and the square root calculator handles the halved exponents that roots produce.