Standard Form

Standard form is a convenient way to represent very small or very large numbers. A number written in standard form is expressed as A\times 10^{n}, where A\geq 1 but <10, and n is an integer.

In other words, 1\le A<10.

For example, the speed of light is approximately 299,793 , \text{km} , \text{s}^{-1}. In standard form, this is 2.99793\times 10^{5} , \text{km} , \text{s}^{-1} or, when rounded to three significant figures, 3\times 10^{5} , \text{km} , \text{s}^{-1}.

Let’s look at some more examples:

Examples of Standard Form

Example 1

Rewrite the following numbers in standard form:

a) 125,600,000,000

b) 1,300

c) 0.000001256

d) 0.0013

The standard form of these numbers is:

a) 1.256\times 10^{11}1.256\times 10^{11}

b) 1.3\times 10^{3}1.3\times 10^{3}

c) 1.256\times 10^{-6}1.256\times 10^{-6}

d) 1.3\times 10^{-3}1.3\times 10^{-3}

In these examples, AA is a number such that 1\le A<101\le A<10, and the power nn is an integer. Large numbers greater than 10 have a positive nn (e.g., a and b), while small numbers less than 1 have a negative nn (e.g., c and d).

Example 2

Write down the following numbers in standard form:

i) 14,500,000

ii) 0.00000145

iii) 145

iv) 14.5

v) 0.145

In standard form, these numbers are:

i) 1.45\times 10^{7}1.45\times 10^{7}

ii) 1.45\times 10^{-6}1.45\times 10^{-6}

iii) 1.45\times 10^{2}1.45\times 10^{2}

iv) 1.45\times 10^{1}1.45\times 10^{1}

v) 1.45\times 10^{-1}1.45\times 10^{-1}

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