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 a vacuum, $$c = 300,000,000$$ meters per second (approximately), can be written more compactly as $$c = 3 \times {10^8}$$ meters 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 the 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 point 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 $$1,087,000,000,000,000,000,000$$ cubic meters.
2. The earth rotates about its axis with an angular speed of approximately $$0.00417$$ degrees 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 $$4.17 \times {10^{ – 3}}$$ degrees 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 divide zeros into groups of three.)

Names and scientific prefixes for some integer powers of $$10$$ are listed in the table below.