How to convert exponential notation to standard form
Move the mantissa's decimal point by however many places the exponent says Example: 5 × 10⁻⁴ — move the point 4 places left — 0.0005 A positive exponent moves the decimal point right (making the number larger); a negative exponent moves it left (making the number smaller). 6.022 × 10²³ (roughly Avogadro's number) means moving the point 23 places right, landing on 602,200,000,000,000,000,000,000 — a number too large to type comfortably in standard form, which is exactly why exponential notation exists for values at that scale in the first place.
This works identically whether the mantissa follows scientific notation's convention (always between 1 and 10) or engineering notation's wider one (up to just under 1,000, so the exponent lands on a multiple of 3). The math is the same either way — mantissa × 10exponent — so there's nothing to detect or switch between; just enter whatever mantissa and exponent you have.
Notation equivalents
| Notation | Standard Form |
|---|---|
| 5 × 10⁻⁴ | 0.0005 |
| 2.5 × 10⁻⁵ | 0.000025 |
| 5 × 10⁻¹ | 0.5 |
| 3 × 10⁰ | 3 |
| 1.5 × 10² | 150 |
| 15 × 10³ (engineering) | 15,000 |
| 4.5 × 10³ | 4,500 |
| 9.3 × 10⁵ | 93,000,000 |
| 6.022 × 10²³ | 602,200,000,000,000,000,000,000 |
Mantissa rules: is what you were given valid?
The number in front of the power of ten is the mantissa (also called the significand or coefficient). Each notation puts a different constraint on it, and the exponent follows from that:
| Notation | Mantissa | Exponent |
|---|---|---|
| Scientific | 1 ≤ |m| < 10 | Any integer |
| Engineering | 1 ≤ |m| < 1000 | A multiple of 3 |
Engineering notation gives up the tidy single digit in order to keep the exponent on multiples of three, so it lines up with the SI prefixes — 10³ is kilo, 10⁻⁶ is micro, 10⁹ is giga. That is the whole reason it exists.
The same numbers in both
| Standard form | Scientific | Engineering |
|---|---|---|
| 47,500 | 4.75 × 10⁴ | 47.5 × 10³ |
| 0.00085 | 8.5 × 10⁻⁴ | 850 × 10⁻⁶ |
| 0.5 | 5 × 10⁻¹ | 500 × 10⁻³ |
| 602,000,000,000,000,000,000,000 | 6.02 × 10²³ | 602 × 10²¹ |
Note the last row: 6.02 × 10²³ is an approximation of the Avogadro constant in scientific notation, and 602 × 10²¹ is the same value written so the exponent is a multiple of three. Neither notation is more precise than the other; both preserve the same three significant figures.
Forms you will meet that are not valid
If a value you have been given does not fit the table above, it is still a number — it is simply not in normalised form, and the converter above will accept it and give you the standard form regardless.
| As written | Problem | Normalised |
|---|---|---|
| 0.5 × 10⁴ | Mantissa below 1 | 5 × 10³ |
| 12.3 × 10⁵ | Mantissa 10 or more | 1.23 × 10⁶ |
| 45 × 10⁴ | Exponent not a multiple of 3, so not engineering either | 450 × 10³ |
| 1.5 × 10⁴ | Valid scientific, but not engineering | 15 × 10³ |
Two edge cases worth knowing. Zero has no normalised form in either notation, because no mantissa of 1 or more multiplied by a power of ten gives zero — it is written as plain 0. And for negative values the sign belongs to the mantissa, not the exponent: −0.00042 is −4.2 × 10⁻⁴, where the minus sign in the exponent and the one on the mantissa mean entirely different things.
The mantissa also carries all the significant figures. Writing 4.75 × 10⁴ claims three of them; writing 4.750 × 10⁴ claims four. The exponent does not determine the number of significant figures, so changing notation does not by itself change the precision represented.
Looking to go the other direction?
To convert a standard number into scientific notation, use the Scientific Notation Converter.
Frequently asked questions
What are the mantissa rules for scientific and engineering notation?
In scientific notation the mantissa must be at least 1 and less than 10, with any integer exponent. In engineering notation it must be at least 1 and less than 1000, with the exponent restricted to multiples of 3 so it matches the SI prefixes. The same value is 4.75 × 10⁴ in one and 47.5 × 10³ in the other.
Is 0.5 × 10⁴ wrong?
It is a perfectly good number — 5000 — but it is not normalised scientific notation because the mantissa is below 1. The normalised form is 5 × 10³. The converter above accepts either and gives the same standard form.
Can engineering notation have a mantissa greater than 10?
Yes. Unlike scientific notation, engineering notation allows the mantissa to range from 1 up to, but not including, 1000. For example, 47.5 × 10³ is valid engineering notation, while 47.5 × 10³ is not normalised scientific notation.
How do I convert scientific notation to decimal?
Move the mantissa's decimal point right for a positive exponent, or left for a negative one, by however many places the exponent specifies. 5 × 10⁻⁴ becomes 0.0005 by moving the point 4 places left.
What is 5 x 10^-4 in standard form?
0.0005.
What does a negative exponent mean in scientific notation?
The number is smaller than 1. A negative exponent moves the decimal point to the left, producing more leading zeros the more negative it is.