- A real number x, times a power of 10 (10n) is written in Scientific Notation if the absolute value of x is greater than or equal to 1, but less than 10, i.e., the absolute value of x is in the interval [1,10)
- The exponent n is an integer
- In all cases, there is only one digit to the left of the decimal point and that digit must be between 1 and 9 inclusive (not 0)
- Multiplying a number by a power of 10 or 10n shifts the decimal point to the right n times if n is a positive integer
- Example: In Scientific Notation, 523 = 5.23 x 102
- Multiplying a number by 10n shifts the decimal point to the left n times if n is a negative integer
- Example: 0.0035 = 3.5 x 10-3
- Application: Expressing multi-digit numbers in a more concise and usable form
- Example: The distance from the Earth to the Sun is 92,956,000 miles, which equals 9.2956 x 107 miles
- Note: The multi-digit numbers could be very small since very small numbers contain multiple digits
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, or standard form in the United Kingdom. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.
| Decimal notation | Scientific notation |
|---|---|
| 2 | 2×100 |
| 300 | 3×102 |
| 4321.768 | 4.321768×103 |
| −53000 | −5.3×104 |
| 6720000000 | 6.72×109 |
| 0.2 | 2×10−1 |
| 987 | 9.87×102 |
| 0.00000000751 | 7.51×10−9 |
In scientific notation, nonzero numbers are written in the form
or m times ten raised to the power of n, where n is an integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m, as in ordinary decimal notation. In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand m is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation.