Scientific Notation Calculator

Scientific notation is a way of expressing very large or very small numbers concisely, written as a coefficient (between 1 and 10) multiplied by a power of 10. For example, the speed of light (299,792,458 m/s) is written as 2.998 × 10⁸, and a hydrogen atom (0.000000000106 m) is written as 1.06 × 10⁻¹⁰. This notation is standard in science, engineering, and computing to avoid writing out long strings of zeros.

This calculator does the conversion both ways and removes the chance of miscounting zeros. Type any number — a tiny decimal or a number in the millions — and it returns the standard scientific-notation form, splitting out the coefficient and the exponent. A second tool on the page multiplies two numbers that are already in scientific notation. It is built for school and college students in physics and chemistry, for anyone reading data-sheet values like 4.7 × 10⁻⁶ F, and for programmers who meet the same idea as "E notation" in their code.

Format

The number a is the coefficient (also called the mantissa or significand) and must be at least 1 and less than 10 — exactly one non-zero digit sits to the left of the decimal point. The number n is the exponent, and it tells you the order of magnitude: how many places the decimal point has been shifted. A positive exponent means a large number, a negative exponent means a small one, and an exponent of 0 means the number is already between 1 and 10. Writing the same value as 34 × 10⁵ is technically equal but is not proper scientific notation, because the coefficient 34 is not between 1 and 10.

Converting To Scientific Notation

Large numbers: count places moved left → positive exponent. 5,000 = 5 × 10³

Small numbers: count places moved right → negative exponent. 0.005 = 5 × 10⁻³

The recipe is the same in both directions: slide the decimal point until exactly one non-zero digit remains in front of it, then count how many places you moved. Moving the point to the left(you started with a big number) gives a positive exponent equal to that count; moving it to theright (you started with a small number) gives a negative exponent. To go back to ordinary (standard) form you simply reverse the move: a positive exponent shifts the point right by that many places, padding with zeros, and a negative exponent shifts it left.

Worked Example

Convert 0.00042 to scientific notation. Start at the decimal point and move it right until a single non-zero digit (the 4) sits in front: 0.00042 → 4.2, a move of four places to the right. Because the move was to the right, the exponent is negative four, giving 4.2 × 10⁻⁴. To check it, read the notation backwards: a −4 exponent means shift the point four places left from 4.2, which rebuilds 0.00042. Now a large example: 73,500 → move the point four places left to get 7.35, so the answer is 7.35 × 10⁴. Type either value into the converter above to see the coefficient and exponent broken out step by step.

Significant Figures

One quiet advantage of scientific notation is that it makes significant figures unambiguous. Every digit you write in the coefficient counts as significant, so 3.4 × 10⁵ shows 2 significant figures while 3.40 × 10⁵ deliberately shows 3 — the trailing zero now carries real meaning about precision. In plain form the number 3,400 is ambiguous about whether those trailing zeros are measured or merely placeholders; scientific notation settles the question by putting only the meaningful digits in the coefficient. This is why lab reports and exam answers in physics and chemistry are expected to use it.

Multiplying in Scientific Notation

To multiply, handle the two parts separately: multiply the coefficients together and add the exponents. For example (3 × 10⁴) × (2 × 10³) = (3 × 2) × 10⁽⁴⁺³⁾ = 6 × 10⁷. If the resulting coefficient lands outside the 1-to-10 range you tidy it up by one more shift — for instance 15 × 10⁵ becomes 1.5 × 10⁶, moving one factor of ten out of the coefficient and into the exponent. Division works the mirror image: divide the coefficients and subtract the exponents. The multiplication tool at the bottom of this page does the arithmetic and the tidy-up for you.

Tips and Common Mistakes

The most frequent slip is getting the sign of the exponent backwards — small numbers (less than 1) always take a negative exponent, large numbers take a positive one. Another is leaving the coefficient out of range, such as writing 0.5 × 10³ or 12 × 10²; normalise so there is exactly one non-zero digit before the decimal. When multiplying, remember you add exponents, not multiply them, and that you only multiply the coefficients. Finally, in E notation the letter e simply replaces "× 10^", so 1.5e6 and 1.5 × 10⁶ are the same value — do not confuse this e with the mathematical constant 2.718.