Learning Objectives#

By the end of this section, you will be able to:

• Identify numbers written in scientific notation

• Convert numbers between scientific notation and standard notation

Definition#

Scientific notation is a specific format used by engineers and scientists to display a number. The rules that define the scientific notation number format are below:

• Is of the form $m\times {10}^n$

• all digits in the significand $m$ are significant

• The significand $m$ is restricted such that $1\le m<10$

• $n$ is an integer, positive or negative

Examples#

• $6.022\times {10}^{23}$ atoms

• $2.040\times {10}^4$ meters

• $9.000\times {10}^{-6}$ grams

All of the examples above show values writen in scientific notation and contain 4 significiant figures.

Scientific notation is particularly useful when the numbers are very large or very small. For example the number of atoms in a mole is 602200000000000000000000 atoms/mole or the Planck constant $h$ which is 0.000000000000000000000000000000000663 J$\cdot$sec. It is cumbersome to write all the zeros in both of these cases.

Scientific notation is an alternative, compact representation of these numbers. For scientific notation, the factor ${10}^n$ indicates the power of ten to multiply the coefficient by to convert back to decimal form.

Image Credit: EGR1010: Introduction to Engineering for Engineers and Scientists, Licensed CC BY-NC-SA 4.0

This is equivalent to moving the decimal in the coefficient fifteen places to the right.

A negative exponent indicates that the number is very small:

Image Credit: EGR1010: Introduction to Engineering for Engineers and Scientists, Licensed CC BY-NC-SA 4.0

This is equivalent to moving the decimal in the coefficient eleven places to the left.

Let’s re-write Planck’s constant and the number of atoms per mole in scientific notation:

Summary#

Scientific notation is useful when measured or calculated values are very large or very small. Scientific notation is of the form:

$m\times {10}^n$

where $1\le m<10$ and $n$ is an integer.

Measured or calculated values can be converted from standard notation into scientific notation by converting the location of a decimal point into an exponent. To go from scientific notation to standard notation, convert the exponent into a located decimal point.