Introduction to Scientific Notation

In applied mathematics, very large and very small numbers are written in compact form by using integer powers of 10. For instance, the speed of light in vacuum, c = 300,000,000 meters per second (approximately), can be written more compactly as c = 3 \times {10^8} meter per second.

More generally, a real number x is said to be expressed in scientific notation if it is written in the form x = \pm p \times {10^n},

Where n is an integer and p is a number greater than or equal to 1, but less than 10. The integer n is called the characteristic of x, the number p is called the mantissa of x, and factor {10^n} is called the order of magnitude of x.

To change a number from ordinary decimal form to scientific notation, move the decimal point to obtain a number between 1 and 10(one digit in front of the decimal point) and multiply by {10^n} or by{10^{ - n}}, where n is the number of places  the decimal points was moved to the left or to the right, respectively. Final zeros after the decimal point can be dropped unless it is necessary to retain them to indicate the accuracy of an approximation.

Example:

Rewrite each statement so that all numbers are expressed in scientific notation.

  1. The volume of the earth is approximately 1,087,000,000,000,000,000,000cubic meters.
  2. The earth rotates about its axis with an angular speed of approximately 0.00417 degree per second.

Solution:
           
(a) We move the decimal point 21 places to the left 1.087000000000000000000
to obtain a number between 1 and 10 and multiply by {10^{21}}, so that 1,087,000,000,000,000,000,000  = 1.087 \times {10^{21}}.
Thus, the volume of the earth is approximately 1.087 \times {10^{21}}cubic meters.

(b) We move the decimal point three places to the right 0004.17 to obtain a number between 1 and 10 and multiply by {10^{ - 3}}, so that 0.00417 = 4.17 \times {10^{ - 3}}.

Thus, the earth rotates about its axis with an angular speed of approximately 4.17 \times {10^{ - 3}} degree per second.

Example:

Rewrite the following numbers in ordinary decimal form.
            (a) 7.71  \times {10^5}       (b) 6.32  \times {10^{ - 8}}
Solution:
            (a) 7.71 \times {10^5} = 771000. = 771,000
            (b) 6.32  \times {10^{ - 8}} = 0.0000000632 = 0.000,000,0632
(Very small numbers, written in ordinary decimal form, are easier to read if commas are used to set off zeros in groups of three.)

Names and Scientific prefixes for some integer powers of 10 are listed in table.

Power of 10

Names

Prefix

10

Ten

Deka

{10^2}

Hundred

Hecto

{10^3}

Thousand

Kilo

{10^6}

Million

Mega

{10^9}

Billion

Giga

{10^{12}}

Trillion

Tera

{10^{15}}

Quadrillion

Peta

{10^{ - 1}}

Tenth

Deci

{10^{ - 2}}

Hundredth

Centi

{10^{ - 3}}

Thousandth

Milli

{10^{ - 6}}

Millionth

Micro

{10^{ - 9}}

Billionth

Nano

{10^{ - 12}}

Trillionth

Pico

{10^{ - 15}}

Quadrillionth

Femto