Scientific Notation

Scientific Notation: A Matter of Convenience

Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers. In scientific notation all numbers are written in the form of $a\cdot 10^{b}$  ($a$ multiplied by ten raised to the power of $b$), where the exponent $b$ is an integer, and the coefficient $a$ is any real number.

Scientific Notation

There are three parts to writing a number in scientific notation: the coefficient, the base, and the exponent.

Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. Thomas Young's discovery that light was a wave preceded the use of scientific notation, and he was obliged to write that the time required for one vibration of the wave was "$\displaystyle \frac{1}{500}$ of a millionth of a millionth of a second"; an inconvenient way of expressing the point. Scientific notation is a less awkward and wordy way to write very large and very small numbers such as these.

A Simple System

Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten.

For instance, $32 = 3.2\cdot 10^{1}$

$320 = 3.2\cdot 10^{2}$

$3200 = 3.2\cdot 10^{3}$, and so forth...

Each number is ten times bigger than the previous one. Since $10^{1}$ is ten times smaller than $10^{2}$, it makes sense to use the notation $10^{0}$ to stand for one, the number that is in turn ten times smaller than $10^{1}$. Continuing on, we can write $10^{-1}$ to stand for 0.1, the number ten times smaller than $10^{0}$. Negative exponents are used for small numbers:

$3.2=3.2\cdot 10^{0}$

$0.32=3.2\cdot 10^{-1}$

$0.032=3.2\cdot 10^{-2}$

Scientific notation displayed calculators can take other shortened forms that mean the same thing. For example, $3.2\cdot 10^{6}$ (written notation) is the same as $3.2\text{E+6}$ (notation on some calculators) and $3.2^{6}$ (notation on some other calculators).