Scientific Notation Calculator

This page holds two tools. The converter turns any number into scientific notation, E-notation, engineering notation, and plain decimal form, so 1568938 becomes 1.568938 × 10⁶. The calculator below it takes two numbers already written in scientific notation and adds, subtracts, multiplies, divides, or raises them, showing the working. Both accept input as plain digits, as 2.3e11, or as 3.5x10^-12.

What scientific notation is

Scientific notation writes any number as a coefficient multiplied by a power of ten, in the form b × 10ⁿ. The coefficient b, also called the significand or mantissa, is at least 1 and less than 10. The exponent n is a whole number, and it is what people mean by the order of magnitude.

The point of it is compactness and comparability. Avogadro's number written out is 602,214,076,000,000,000,000,000; in scientific notation it is 6.02214076 × 10²³. The mass of an electron is 9.109 × 10⁻³¹ kilograms, which nobody would want to write as a decimal. More usefully, two numbers in this form can be compared at a glance by their exponents alone.

• 5 is 5 × 10⁰
• 700 is 7 × 10²
• 1,000,000 is 1 × 10⁶
• 0.0004212 is 4.212 × 10⁻⁴
• -5,000,000,000 is -5 × 10⁹

Converting a number by hand

Move the decimal point until exactly one non-zero digit sits to its left, then count how far it travelled. Moving it left gives a positive exponent, moving it right gives a negative one.

Take 1568938. The decimal point starts at the right-hand end and has to move six places left to sit after the 1, giving 1.568938 × 10⁶. Take 0.0004212. The point moves four places right to sit after the 4, so the exponent is negative: 4.212 × 10⁻⁴. The sign of the exponent tells you which side of 1 the number lives on, and nothing else.

Going the other way, a positive exponent means shifting the point right and padding with zeros, and a negative exponent means shifting it left.

Adding and subtracting

Addition and subtraction are the awkward ones, because the powers of ten have to match before you can do anything. Convert the smaller number so it shares the larger exponent, add or subtract the coefficients, then renormalise.

Take 1.23 × 10⁷ + 3.45 × 10². The larger exponent is 7, so rewrite the second number against it: 3.45 × 10² is 0.0000345 × 10⁷. Now the coefficients can be added directly: 1.23 + 0.0000345 = 1.2300345, giving 1.2300345 × 10⁷, or 12,300,345.

A simpler case shows the shape more clearly. 4.5 × 10⁵ + 2.3 × 10⁴ becomes 4.5 × 10⁵ + 0.23 × 10⁵, which is 4.73 × 10⁵. Notice what happens when the exponents are far apart: adding 3.45 × 10² to 1.23 × 10⁷ changes the answer only in the fifth decimal place of the coefficient. That is not a rounding artefact, it is the correct result, and it is why quantities of wildly different magnitude often make one term irrelevant.

Multiplying and dividing

These are the easy ones, because the two parts separate cleanly. To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents.

(3.2 × 10⁴) × (2.5 × 10⁻⁷) gives 3.2 × 2.5 = 8 for the coefficients and 4 + (-7) = -3 for the exponents, so the answer is 8 × 10⁻³, or 0.008. Division works the same way in reverse: (6.4 × 10⁸) ÷ (1.6 × 10³) gives 6.4 ÷ 1.6 = 4 and 8 - 3 = 5, so 4 × 10⁵.

Raising to a power follows from the same logic when the exponent is a whole number: square a number and you square the coefficient and double the exponent, so (1.23 × 10⁷)² is 1.5129 × 10¹⁴. For roots, halve the exponent, shifting one power across first if it is odd so you are not left with a fraction.

Renormalising the answer

Arithmetic often leaves a coefficient outside the 1-to-10 range, and the result is not in scientific notation until you fix that. The rule is a trade: every place you move the decimal point one way, you change the exponent one step the other way.

If a division leaves 0.179 × 10⁻², the coefficient is below 1, so shift the point one place right and drop the exponent by one: 1.79 × 10⁻³. If a multiplication leaves 34 × 10⁻³, the coefficient is above 10, so shift the point one place left and raise the exponent: 3.4 × 10⁻². Both numbers were already correct; they simply were not in standard form. This calculator normalises automatically, which is why the coefficient in the result is always between 1 and 10.

Engineering notation

Engineering notation is scientific notation with one extra rule: the exponent must be a multiple of three. That restriction exists so the exponents line up with SI prefixes, which makes the number readable aloud.

The trade-off is that the coefficient can now be anything from 1 up to 1000. So 1.234 × 10⁸ in scientific notation becomes 123.4 × 10⁶ in engineering notation, which reads as 123.4 mega-something. Likewise 4.212 × 10⁻⁴ becomes 421.2 × 10⁻⁶, or 421.2 micro-units.

The prefixes worth knowing run in steps of three in both directions: kilo is 10³, mega 10⁶, giga 10⁹, and tera 10¹²; going down, milli is 10⁻³, micro 10⁻⁶, nano 10⁻⁹, and pico 10⁻¹². This is why component values, data rates, and frequencies are almost always quoted in engineering form rather than strict scientific form.

E-notation

E-notation replaces the "× 10" with the letter e, so 1.568938 × 10⁶ becomes 1.568938e6. It exists because most keyboards, calculator displays, and programming languages cannot render a raised exponent.

• 5 × 10⁰ is 5e0
• 7 × 10² is 7e2
• 1 × 10⁶ is 1e6
• 4.212 × 10⁻⁴ is 4.212e-4
• -5 × 10⁹ is -5e9

Capital E and lowercase e mean the same thing, and this calculator outputs lowercase. One warning: the e in E-notation has nothing to do with Euler's number, 2.71828, despite sharing the letter. Spreadsheets and programming languages accept E-notation as typed input, which makes it the practical format for moving a number between this page and a spreadsheet.

Significant figures

Scientific notation solves an ambiguity that plain decimals cannot. Written as 1500, a measurement could carry two, three, or four significant figures, and the notation gives no way to tell. Written as 1.5 × 10³ it is unambiguously two; as 1.500 × 10³ it is four.

That matters whenever a figure came from a measurement rather than a count, because the trailing digits carry a claim about precision. The general rule for calculations is that a result should not be quoted to more significant figures than the least precise input, so multiplying a 3-significant-figure value by a 5-significant-figure one gives an answer good to three. This calculator does not enforce that for you, so round the final answer yourself.

How this page works

The converter parses your input, rewrites it with exactly one digit before the decimal point, and reports all four forms alongside the order of magnitude. The step-by-step panel shows where the decimal point moved and why the exponent came out as it did.

The calculator takes each number as a coefficient and an exponent, applies the rule for the operation you choose, and normalises the result. The precision field controls how many digits appear after the decimal point in the answer. Results are computed in standard double-precision arithmetic, which carries about 15 to 17 significant digits, so answers are exact well beyond ordinary needs but not infinitely so. For related work, the Exponent Calculator handles powers on their own, the Root Calculator covers nth roots, the Log Calculator converts between orders of magnitude, and the Scientific Calculator evaluates full expressions.