### 2.5 - Scientific Notation

In scientific applications very large or very small numbers are encountered all the time.
An example of a very large number is the speed of light (300000000 meters per
second) and an example of a very small number is the charge of an electron
(0.00000000000000000016 coulombs). To make these numbers easier to read
they are usually expressed in **scientific notation**, like this:
300000000 = 3 × 10^{ 8}

0.00000000000000000016 = 1.6 × 10^{ − 19}

The rule for writing a number in scientific notation is to write it the
product of two numbers. The first number, called the mantissa, must
have its absolute value between 1 and 10,
and the second number must be written as 10 raised to an integer power. (If
the mantissa is exactly 1 then it may be omitted.) Calculators
often replace the symbols “× 10 ” by the symbol E to save
space on the display screen, like this:
300000000 = 3 E 8

0.00000000000000000016 = 1.6 E − 19

**Notes: **
- A common mistake is the think that 3 E 8 means 3
^{8}.
This is not correct. 3 E 8 means 3 × 10^{8}.

- The following forms all have the same value but only the first
one is scientific notation. To change from one form to the next
we multiply the mantissa by 10 and subtract 1 from the power of 10
(effectively dividing the second number by 10). The result is that
the value of the product remains unchanged:

3 × 10^{8} = 30 × 10^{7} = 300 × 10^{6} = 3000 × 10^{5}

- A notation similar to scientific notation is
**engineering notation**.
In this notation the mantissa is multiplied or divided by enough powers of 10
so that the power of 10 in the second number can be a multiple of 3.
(Engineers like to count in thousands, millions, billions, etc.)
For example 3 × 10^{8} will be written either as 0.3 × 10^{9}
or as 300 × 10^{6}.

- The Algebra Coach can accept any of these forms as input:

300000000 or 3 E 8 or 3 * 10 ^ 8

### Multiplying or dividing numbers in scientific notation

To multiply or divide numbers that are expressed in scientific notation,
simply multiply or divide the mantissas to produce the new mantissa
and combine the powers of 10 together using the
properties of exponents
to produce the new power of 10. If the new mantissa is
not between 1 and 10 then adjust it and the power of 10 as in the
second note above.

Here are some examples:
(1.2 × 10^{8 }) (2.3 × 10^{10 })
= 1.2 × 2.3 × 10^{8} × 10^{10}

= 2.76 × 10^{18}

(6 × 10^{8 }) (7 × 10^{10 })
= 42 × 10^{18}
← mantissa is too big

= 4.2 × 10^{19}
so divide it by 10 and add 1 to the power of 10

= 0.5 × 10^{ − 2}
← mantissa is too small

= 5 × 10^{ − 3}
so multiply it by 10 and subtract 1 from the power of 10

### Adding and subtracting numbers in scientific notation

In order to add or subtract numbers expressed in scientific notation
the power of 10 factor must be the same for all of the numbers.
To accomplish this use the above method of multiplying
or dividing the mantissa by a power
of 10 and adjusting the power of 10 factor accordingly.
Here are some examples:
8.4 × 10^{ 10} + 5.3 × 10^{ 10}
← powers of 10 are the same so just go ahead and add mantissas
= 13.7 × 10^{ 10}
← mantissa is too big so adjust it

= 1.37 × 10^{11}

8 × 10^{ 6} + 5.1 × 10^{ 7}
← the powers of 10 are different; adjust the first number
= 0.8 × 10^{ 7} + 5.1 × 10^{ 7}
← powers of 10 the same; now add mantissas

= 5.9 × 10^{7}