Concept

Scientific Notation

Scientific notation is a compact way of writing very large or very small numbers. A number written in scientific notation is expressed as a product of two numbers.

a \times 1 0^{b}

In this form, the first factor is greater than or equal to

1

and less than

10 .

In other words, it needs to be in the interval

[1, 10) .

The second factor is a power of

10

where

b

is an integer. For example, the number

4

million can be rewritten as the product of

4

and a multiple of

10 .

Then, the multiple of

10

is rewritten as a base

10

power.

4000000 = 4 \times 1000000 = 4 \times 1 0^{6}

Very small decimal numbers can also be written in scientific notation. Consider a number where there are many zeros before the significant figures. Take as

0.000342

as an example.

0.000342 = 3.42 \div 10000 = 3.42 \times 1 0^{- 4}

In such cases, numbers are expressed as a division by a multiple of

10 .

Division by a multiple of

10

is equivalent to multiplication by a base

10

power with a negative exponent. Consider a few more examples of numbers written in scientific notation.

Decimal Form	Written as a Product or Division Expression	Scientific Notation
$4505$	$4.505 \times 1000$	$4.505 \times 1 0^{3}$
$8320000$	$8.32 \times 1000000$	$8.32 \times 1 0^{6}$
$0.0005$	$5 \div 10000$	$5 \times 1 0^{- 4}$
$0.0521$	$5.21 \div 100$	$5.21 \times 1 0^{- 2}$

Intuitive Method: Rewriting a Number in Scientific Notation

An intuitive method to rewrite a number into scientific notation is to count the number of places the decimal needs to move. Consider a number greater than

10 .

The decimal would move from right to left to make the number less than

10

but still greater than

1 .

The number of places the decimal moved indicates the positive exponent to be used for the base

10

power.

Similarly, for numbers less than

1,

such as

0.000022,

the decimal will move from left to right to make the number greater than or equal to

1

and less than

10 .

In this case, the number of places moved indicates the negative exponent to be used for the base

10

power.

Scientific notation is not only a convenient way to express cumbersome numbers. It also eases the comparison of numerical order of magnitude. For example, it may be difficult to determine how much larger

237000000

is compared to

4530000 .

However, it is easier to see that

2.37 \times 1 0^{8}

and

4.5 \times 1 0^{6}

differ by a factor of about

1 0^{2} = 100 .

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Intuitive Method: Rewriting a Number in Scientific Notation