# Scientific notation

Scientific notation (also referred to as exponential notation) is a concise way of expressing real numbers with many digits that would otherwise be inconvenient to represent in decimal notation. This is accomplished by using integer powers of the base used for the positional notation in use.

A number $$\alpha$$ is written in scientific notation if it occurs in the form:

$\alpha = k \cdot 10^n$

where $$k$$ is a decimal number greater than or equal to 1 and less than 10 and $$n$$ is an integer.

For example: the numbers 3,5 · 107 and 8,9 · 10−5 are written in scientific notation, while the numbers 0,5 · 103 and 10,3 · 10−8 are not written in scientific notation because the number in front of the power of 10 in the first case is 0.5 which is less than 1, in the second case is 10.3 which is greater than 10.

## How to transform a number into scientific notation

Consider the measure of the diameter of the red blood cell, or 0.000007 m. To express this measure in scientific notation, we need only consider its generating fraction, ie:

$0.000007\;\textrm{m} = 7 \cdot 1/10000\;\textrm{m} = 7 \cdot 10^{-6}\;\textrm{m}$

Note: To “small” numbers, corresponds a power of ten with negative exponent; to “large” numbers, corresponds a power of ten with positive exponent.

Procedure: write a positive decimal number a in scientific notation, if a > 1:

1. you divide the decimal number by a power of 10 so that you have a decimal number including greater than or equal to 1 and less than 10. To find the power of 10 by which to divide the number you must count the significant digits of the number before any decimal point and remove 1;
2. To write the number a in scientific notation, multiply the number found in the previous step by the power of 10.

To write a positive decimal number a in scientific notation, if 0 < a < 1:

1. you multiply the decimal number by an appropriate power of 10 so that you get a number greater than or equal to 1 and less than 10. To find the power of 10 you must count the zeros that are between the decimal point and the first significant digit of the number and add 1;
2. to write the number a in scientific notation it is necessary to multiply the number obtained in the previous step by the same power of 10 used but taken with negative exponent.

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