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# Scientific notation

Module by: Pinelands High School. E-mail the author

## Scientific Notation

Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

a × 10 b a × 10 b a times 10^{b}
(1)

(" a times ten to the power of b "), where the exponent b is an integer , and the coefficient a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm ). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

### What is the use of scientific notation in every day life?

Scientific notation is needed any time you need to express a number that is very big or very small. Suppose for example you wanted to figure out how many drops of water were in a river 12 km long, 270 m wide, and 38 m deep (assuming one drop is one millilitre). It's much more compact and meaningful to write the answer as roughly 1,23 × 10 14 1,23 × 10 14 1,23 times 10^{-14} than it is to write 123120000000000. For one thing, the scientific notation is easier to read, and makes it much easier to tell at a glance what the order of magnitude is (rather than counting zeros).

For another, most of the digits in 123120000000000 are completely meaningless (unless your measurements were very precise). For instance, if the exact river length were really 12.123123 km (we just measured it to the nearest kilometre), then correct number of drops would be 124383242000000, and after the first three digits our result of 123120000000000 is quite inaccurate. So it's better to use a notation (like scientific notation) in which you can suppress the inaccurate digits

### Converting numbers to scientific notation

Count the number of places to move the decimal point to the right or left and write the number like this:

The number 1,350,000,000 can be written as

1,35 × 10 9 1,35 × 10 9 1,35 times 10^9
(2)

The number 0.000000000000017 can be written as

1,7 × 10 14 1,7 × 10 14 1,7 times 10^{-14}
(3)

### Dividing numbers written in scientific notation

State the problem: 7,5×1011÷4,5×10+207,5×1011÷4,5×10+20{7,5 times 10^{-11}} div {4,5 times 10^{+20}}

Group the factors: (7,5÷4,5)×(1011÷10+20)(7,5÷4,5)×(1011÷10+20)(7,5 div 4,5) times ( 10^{-11} div 10^{+20})

Subtract the exponents: 1,67×10(11(+20))1,67×10(11(+20))1,67 times 10^(-11 - (+ 20) )

### Multiplying numbers written in scientific notation

State Problem: (1,5×1011)×(4,5×10+20)(1,5×1011)×(4,5×10+20)(1,5 times 10^{-11}) times (4,5 times 10^{+20})

Group the factors: (1,5×4,5)×(1011×10+20)(1,5×4,5)×(1011×10+20)(1,5 times 4,5) times (10^{-11} times 10^{+20})

Add the exponents: 6,75×10(11+20)6,75×10(11+20)6,75 times 10^(-11 + 20)

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