Scientific notation calculator that shows the count, not just the answer
Convert a number to scientific notation or expand one back to standard form. Each conversion shows where the decimal point moved, how many places, and why that direction makes the exponent positive or negative, which is the part that trips most people up.
Large exponents are expanded digit by digit rather than through a floating-point value, so 6.02 × 10²³ comes back with every digit intact.
One nonzero digit before the point, and the exponent counts the move
Scientific notation writes every number as a value between 1 and 10 multiplied by a power of ten. The mantissa must have exactly one nonzero digit before the decimal point, so 52 × 10² and 0.52 × 10⁴ are both the right number written the wrong way.
The exponent is a count, not a mystery. It records how many places the decimal point moved and in which direction. Move it left, as you do when shrinking a large number down to a single leading digit, and the exponent is positive. Move it right, as you do when growing a small decimal, and the exponent is negative.
That sign rule catches people out because it feels backwards: a very small number like 0.00052 has a negative exponent even though the point moved to the right. The way to keep it straight is to ask what the power of ten has to do to undo your move. You shrank the number, so the ten has to grow it back.
A large number
93,000,000 = 9.3 × 10⁷
The point moves 7 places left to sit after the 9, so the exponent is +7.
A small number
0.00052 = 5.2 × 10⁻⁴
The point moves 4 places right to sit after the 5, so the exponent is −4.
Already between 1 and 10
7.5 = 7.5 × 10⁰
No movement needed, so the exponent is zero. 10⁰ = 1, which leaves the number unchanged.
Expanding back
6.02 × 10²³
A positive exponent moves the point right, so this becomes 602 followed by 21 zeros.
Converting a number to scientific notation
That digit becomes the one before the decimal point. In 0.00052 it is the 5; in 93,000,000 it is the 9. Leading zeros never count.
0.00052 becomes 5.2 and 93,000,000 becomes 9.3. Trailing zeros in the mantissa are dropped unless they are significant figures you were told to keep.
From 0.00052 to 5.2 the point moved 4 places right. From 93,000,000 to 9.3 it moved 7 places left.
Left means positive, right means negative. So 93,000,000 = 9.3 × 10⁷ and 0.00052 = 5.2 × 10⁻⁴.
Move the point back the number of places the exponent says, in the opposite direction, and you should land on the number you started with. This takes two seconds and catches a wrong sign immediately.
Both directions, with the count shown
A small number
0.00052- The first nonzero digit is 5.
- Placing the point after it gives a mantissa of 5.2.
- The point moved 4 places to the right, so the exponent is −4.
- Check: 5.2 × 10⁻⁴ means moving the point 4 places left from 5.2, which gives 0.00052.
A large number
93,000,000- The first nonzero digit is 9.
- The mantissa is 9.3; the trailing zeros are absorbed into the power of ten.
- The point moved 7 places to the left, so the exponent is +7.
- This is roughly the distance from the Earth to the Sun in miles, which is why the notation exists: 9.3 × 10⁷ is far easier to compare with 1.5 × 10⁸ than the digit strings are.
Expanding back out
6.02 × 10²³- The exponent is positive, so the point moves right.
- It moves 23 places. The mantissa supplies 3 digits, so 21 zeros are added after them.
- The result is 602,000,000,000,000,000,000,000.
- This is Avogadro's number, and writing it out is a fair demonstration of why nobody does.
Engineering notation is not the same thing
Engineering notation follows the same idea but restricts the exponent to multiples of three, so that it lines up with the metric prefixes: kilo at 10³, mega at 10⁶, milli at 10⁻³, micro at 10⁻⁶. In engineering notation 93,000,000 is written 93 × 10⁶, which reads directly as 93 mega.
That is why the mantissa there is allowed to run from 1 up to 1000, unlike scientific notation where it must stay under 10. Neither is more correct; they are optimised for different jobs. Scientific notation is for comparing magnitudes, engineering notation is for reading off units.
Significant figures travel with the mantissa
One quiet advantage of scientific notation is that it makes precision explicit. Written as 9300, a number gives no hint whether the trailing zeros are measured or merely placeholders. Written as 9.3 × 10³ it clearly carries two significant figures, and 9.300 × 10³ clearly carries four.
This is the reason science papers use the notation even for numbers that are not especially large. It is a statement about how much of the number you actually know.
Scientific notation FAQ
How do you write a number in scientific notation?
Move the decimal point so that exactly one nonzero digit sits before it, then multiply by ten raised to the number of places you moved. Moving left gives a positive exponent, moving right gives a negative one. So 93,000,000 becomes 9.3 × 10⁷ and 0.00052 becomes 5.2 × 10⁻⁴.
Why is the exponent negative for a small number?
Because the power of ten has to undo the move you made. Turning 0.00052 into 5.2 makes the number bigger, so the ten must shrink it back, and only a negative exponent does that. The sign records what the power of ten does, not which way your pencil went.
What is 0.00052 in scientific notation?
5.2 × 10⁻⁴. The first nonzero digit is 5, the decimal point moves four places right to sit after it, and the rightward move makes the exponent negative.
How do you convert scientific notation back to a normal number?
Move the decimal point by the number of places the exponent gives: right for a positive exponent, left for a negative one, adding zeros as needed. 6.02 × 10²³ becomes 602 followed by 21 zeros; 5 × 10⁻³ becomes 0.005.
What is the difference between scientific and engineering notation?
Scientific notation keeps the mantissa between 1 and 10 and allows any integer exponent. Engineering notation restricts the exponent to multiples of three so it lines up with metric prefixes, which lets the mantissa run up to 1000. 93,000,000 is 9.3 × 10⁷ scientifically and 93 × 10⁶ in engineering notation.
Does scientific notation show significant figures?
Yes, and that is one of its main uses. 9300 is ambiguous about whether the zeros are measured. 9.3 × 10³ clearly carries two significant figures and 9.300 × 10³ clearly carries four, because only the mantissa's digits count.
Can this calculator expand as well as compact?
Yes. Type a plain number and it converts to scientific notation; type 6.02e23 or 3.2 x 10^4 and it expands back to standard form. Large exponents are expanded digit by digit rather than through a floating-point value, so no digits are lost.
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