Scientific Notation

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Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

$A \times 10^b$

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number, called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

Ordinary decimal notation	Scientific notation (normalised)
300	$\small 3 \times 10^2$
4,000	$\small 4 \times 10^3$
5,720,000,000	$\small 5.720 \times 10^9$
−0.0000000061	$\small -6.1 \times 10^{-9}$

1 Examples
2 Significant Figures
3 Using scientific notation
- 3.1 Converting
- 3.2 Basic operations

Examples

An electron's mass is about 0.00000000000000000000000000000091093822 kg. In scientific notation, this is written as $\small 9.109 \times 10^{-31} \, kg$ .
The Earth's mass is about 5,973,600,000,000,000,000,000,000 kg. In scientific notation, this is written $\small 5.9736 \times 10^24 \, kg$ .
The Earth's circumference is approximately 40,000,000 m. In scientific notation, this is written $\small 4.00 \times 10^7 \, m$ .

Significant Figures

One advantage of scientific notation is that it greatly reduces the ambiguity of number of significant digits. All digits in normalized scientific notation are significant by convention. But in decimal notation any zero or series of zeros next to the decimal point are ambiguous, and may or may not indicate significant figures (when they are they should be underlined to explicity show that they are significant zeros). In decimal notation, zeros next to the decimal point are not necessarily significant numbers. I.e., they may be there only to show where the decimal point is. In scientific notation, however, this ambiguity is resolved, because any zeros shown are considered significant by convention. For example, using scientific notation, the speed of light in SI units is 2.99792458×108 m/s and the inch is 2.54×10−2 m; both numbers are exact by definition of the units "inches" per cm and "meters" in terms of the speed of light.[4] In these cases, all the digits are significant. A single zero or any number of zeros could be added on the right side to show more significant digits, or a single zero with a bar on top could be added to show infinite significant digits (just as in decimal notation).

Using scientific notation

Converting

To convert from ordinary decimal notation to scientific notation, move the decimal separator the desired number of places to the left or right, so that the mantissa will be in the desired range (between 1 and 10 for the normalized form). If you moved the decimal point n places to the left then multiply by 10ⁿ; if you moved the decimal point n places to the right then multiply by 10⁻ⁿ. For example, starting with 1,230,000, move the decimal point six places to the left yielding 1.23, and multiply by 10⁶, to give the result Template:val. Similarly, starting with 0.000 000 456, move the decimal point seven places to the right yielding 4.56, and multiply by 10⁻⁷, to give the result Template:val.

If the decimal separator did not move then the exponent multiplier is logically 10⁰, which is correct since 10⁰ = 1. However, the exponent part "× 10⁰" is normally omitted, so, for example, 1.234 is just written as 1.234 rather than Template:val.

To convert from scientific notation to ordinary decimal notation, take the mantissa and move the decimal separator by the number of places indicated by the exponent — left if the exponent is negative, or right if the exponent is positive. Add leading or trailing zeroes as necessary. For example, given 9.5 × 10¹⁰, move the decimal point ten places to the right to yield 95,000,000,000.

Conversion between different scientific notation representations of the same number is achieved by performing opposite operations of multiplication or division by a power of ten on the mantissa and the exponent parts. The decimal separator in the mantissa is shifted n places to the left (or right), corresponding to division (multiplication) by 10ⁿ, and n is added to (subtracted from) the exponent, corresponding to a cancelling multiplication (division) by 10ⁿ. For example:

$1.234 \times10^3 = 12.34 \times10^2 = 123.4 \times10^1 = 1234.$

Basic operations

Given two numbers in scientific notation,

$x_0=a_0\times10^{b_0}$

$x_1=a_1\times10^{b_1}$

Multiplication and division are performed using the rules for operation with exponential functions:

$x_0 x_1=a_0 a_1\times10^{b_0+b_1}$

$\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}$

some examples are:

$5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2}$

$\frac{2.34\times10^2}{5.67\times10^{-5}} \approx 0.413\times10^{7} = 4.13\times10^6$

Addition and subtraction require the numbers to be represented using the same exponential part, so that the mantissas can be simply added or subtracted. These operations may therefore take two steps to perform. First, if needed, convert one number to a representation with the same exponential part as the other. This is usually done with the one with the smaller exponent. In this example, x₁ is rewritten as:

$x_1 = c \times10^{b_0}$