How to use scientific notation

Scientific notation writes a very large or very small number as a number between 1 and 10 times a power of ten, like 6.022 × 10²³. Here is how to convert both ways, read negative exponents, and multiply, divide, add and subtract without losing track of the zeros.

By the Calculator Den team · Updated 2026-11-06

Scientific notation is a compact way to write numbers that are very large or very small, so you do not have to count long strings of zeros. It writes any number as a coefficient between 1 and 10, times a power of ten, for example 6.022 × 10²³ or 2 × 10⁻⁶. The power of ten tells you the size; the coefficient tells you the precise value. It is the standard way numbers appear in science and on calculators, and once you know the handful of rules, it is faster to work with than the written-out form. This guide covers converting both ways, negative exponents, and the arithmetic, with worked numbers throughout.

The one-digit rule

The whole convention rests on one rule: the coefficient has exactly one non-zero digit before the decimal point, so it is always at least 1 and less than 10. That is why 6.022 × 10²³ is correct scientific notation but 60.22 × 10²² and 0.6022 × 10²⁴ are not, even though all three equal the same number. The second two are sometimes called "not normalized": they are valid arithmetic but not standard form. Getting the coefficient into the 1-to-10 range is what the conversion steps below are really doing.

How to convert a large number into scientific notation

Put the decimal point after the first non-zero digit, then count how many places you moved it. That count is the exponent, and for a number bigger than 1 the exponent is positive.

Take 4,500. The first digit is 4, so the coefficient is 4.5, and you moved the decimal point 3 places to the left (from after the last zero to after the 4), so 4,500 = 4.5 × 10³. Take the speed of light, about 299,792,458 metres per second: to one or two digits that is roughly 3.0 × 10⁸, because the decimal moves 8 places. The exponent is simply how many powers of ten the coefficient has to be scaled up by. Writing it as 3.0 × 10⁸ rather than 3 × 10⁸ is a choice about significant figures: the coefficient carries the precision you want to keep, while the exponent only fixes the size. Avogadro's number, about 6.022 × 10²³, keeps four figures the same way.

Negative exponents: very small numbers

For a number smaller than 1, the same method applies but you move the decimal point to the right, and the exponent comes out negative. The exponent counts how far below 1 the number sits.

Take 0.00062. Move the decimal point right until it sits after the first non-zero digit (the 6): that is 4 places, giving a coefficient of 6.2, so 0.00062 = 6.2 × 10⁻⁴. A bacterium about 0.000002 metres across is 2 × 10⁻⁶ m. A negative exponent does not mean a negative number: 6.2 × 10⁻⁴ is a small positive number, 0.00062. The minus sign in the exponent only tells you the number is smaller than 1; it does not make the number negative.

How to convert back to an ordinary number

Reading scientific notation back is the reverse: move the decimal point by the exponent, right for a positive exponent and left for a negative one, filling in zeros as needed. So 3.2 × 10⁵ means move the point 5 places right: 320,000. And 7 × 10⁻³ means move it 3 places left: 0.007. A quick sanity check is that a positive exponent should give you a big number and a negative one a small fraction; if it comes out the other way, you moved the point the wrong direction.

Multiplying and dividing in scientific notation

This is where scientific notation earns its keep. To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents.

Multiply: (3 × 10⁴) × (2 × 10³) = (3 × 2) × 10⁴⁺³ = 6 × 10⁷. Divide: (8 × 10⁵) ÷ (2 × 10²) = (8 ÷ 2) × 10⁵⁻² = 4 × 10³. One extra step comes up when the coefficients multiply to 10 or more: (5 × 10⁶) × (4 × 10³) = 20 × 10⁹, but 20 is not between 1 and 10, so renormalize by writing 20 as 2 × 10¹ and adding that extra power: 2 × 10¹⁰. Division can push the coefficient the other way: (2 × 10⁵) ÷ (8 × 10²) = 0.25 × 10³, and since 0.25 is below 1, shift it up one place to 2.5 × 10². Always check the coefficient is back in the 1-to-10 range at the end.

Adding and subtracting (match the exponents first)

Adding and subtracting is the one case where you do not just work on the exponents. The powers of ten must match before you can add the coefficients, exactly like needing a common denominator for fractions. Take 3 × 10⁴ + 2 × 10³. Rewrite the smaller one to share the bigger exponent: 2 × 10³ = 0.2 × 10⁴. Now both are in 10⁴, so add the coefficients: (3 + 0.2) × 10⁴ = 3.2 × 10⁴, which is 32,000. Subtraction works the same way: 5 × 10⁴ − 2 × 10³ becomes 5 × 10⁴ − 0.2 × 10⁴ = 4.8 × 10⁴. If the exponents are far apart, the smaller number barely changes the result, which is itself a useful thing to see at a glance.

E notation on a calculator

Calculators and spreadsheets show scientific notation with an "E" (or "e") instead of "× 10", because there is no superscript on a keypad. So 1.5E9 means 1.5 × 10⁹, which is 1,500,000,000, and 6.02E-23 means 6.02 × 10⁻²³. The number after the E is just the exponent. If a calculator suddenly displays a result as 4.2E-5, it has not broken; it is telling you the answer is 0.000042. This is also how you type scientific notation into most tools: enter 1.5E9 rather than the full string of zeros.

Common mistakes

• A coefficient outside 1 to 10. 42 × 10³ is not standard form. Rewrite it as 4.2 × 10⁴ by shifting one power of ten.
• Thinking a negative exponent means a negative number. 5 × 10⁻³ is 0.005, a small positive number. The sign of the exponent controls size, not whether the number is positive.
• Adding exponents when adding numbers. You add exponents only when multiplying. To add or subtract, match the exponents first, then add the coefficients.
• Forgetting to renormalize. After multiplying or dividing, if the coefficient landed at 10 or more (or below 1), shift the decimal point until it is back between 1 and 10, and change the exponent by the same number of places.

Keep the coefficient between 1 and 10, let the exponent carry the size, add exponents to multiply and subtract them to divide, and match them before adding. When you want it handled instantly, including E-notation input and converting back to a plain number, the free scientific notation calculator does the conversions and arithmetic, and because these numbers turn up in physics, our explainers on Ohm's law and density show them in use.