Method · Basic Math
How to Write a Number in Scientific Notation: Step by Step
There are only three moves: find where the decimal point has to go, count how far it travelled, and let the direction decide the sign.
The three moves
To learn how to write a number in scientific notation, the method is the same for every number, big or small.
- Put a decimal point after the first non-zero digit. That gives the coefficient, a number from 1 up to 10.
- Count how many places the decimal point moved from where it started.
- Write × 10 to that power. Moving the point left gives a positive exponent; moving it right gives a negative one.
The direction rule is the part to memorise: big numbers have positive exponents because the point had to move left to reach the front digit.
A large number
- 45000the number, with its decimal point after the last zero
- 4.5put the point after the first non-zero digit
- 4 placesthe point moved 4 places to the left, past the 4 and three zeros
- 4.5 × 10⁴left means positive, so the exponent is +4
Counting the places rather than the zeros is the habit that prevents most errors here. There are three zeros but the exponent is 4.
A small number
- 0.0000725the number
- 7.25put the point after the first non-zero digit, the 7
- 5 placesthe point moved 5 places to the right
- 7.25 × 10⁻⁵right means negative, so the exponent is −5
Numbers that are already close to 1
A number between 1 and 10 needs no move at all, so its exponent is 0. The number 8 is 8 × 10⁰, and 12.5 is 1.25 × 10¹, since the point moves one place left. A zero exponent is perfectly valid, it simply says that the number is of ordinary size.
Converting back to a standard number
To convert back, run the moves in reverse: read the exponent, then move the decimal point that many places, filling any gaps with zeros.
- 3.8 × 10⁸the number in scientific notation
- positive exponent: move righta positive exponent means a large number
- 380000000move 8 places right, filling the gap with zeros
- 2.5 × 10⁻³the number in scientific notation
- negative exponent: move lefta negative exponent means a small number
- 0.0025move 3 places left, filling with zeros in front
Keeping the significant figures
Scientific notation should carry exactly the digits you were given. 1500 written as 1.5 × 10³ has two significant figures, and written as 1.500 × 10³ it has four. Never drop or add digits when converting: the coefficient holds every digit of the original, just with the decimal point moved.
Slips worth watching for
Three errors account for nearly every wrong answer. The first is moving the point the right distance but the wrong way, which flips the sign of the exponent. The second is stopping one place early or late, leaving a coefficient like 45 or 0.45 instead of 4.5. The third is counting the zeros instead of the places the point travelled.
A useful habit is to say the move aloud in plain words: the point moved four places left, so the number is big, so the exponent is positive four. Naming the direction and the size separately keeps the sign and the count from blurring together, and it takes only a second.
Negative numbers add nothing new. The minus sign stays in front of the coefficient, so −45000 is −4.5 × 10⁴ and −0.0072 is −7.2 × 10⁻³. The sign of the number and the sign of the exponent are separate things, and each answers its own question: is it below zero, and is it small or large.
Checking your answer
Check your answer two ways. First, make sure the coefficient is at least 1 and below 10. Second, move the point back by the exponent and confirm you recover the original number.
The Scientific Notation Calculator on this site does the move on the digits themselves, so even a number with twenty digits keeps every one of them, and it shows each step of the move.
Questions about how to write a number in scientific notation
How do I write 45000 in scientific notation?
4.5 × 10⁴. Put the decimal point after the 4, count that it moved 4 places left, and use a positive exponent.
How do I write 0.0000725 in scientific notation?
7.25 × 10⁻⁵. The point moves 5 places right to sit after the 7, so the exponent is negative.
How do I convert scientific notation to a normal number?
Move the decimal point by the exponent: right for positive, left for negative, filling with zeros. 3.8 × 10⁸ is 380000000.
How do I know if my exponent has the right sign?
Large numbers (10 or more) need a positive exponent, and small numbers (between 0 and 1) need a negative one. A mismatch means the point moved the wrong way.