Scientific notation
Introduction
In engineering calculations numbers are often very small or
very large, for example 0.00000345 and 870,000,000. To avoid
writing lengthy strings of numbers a notation has been developed,
known as scientific notation which enables us to write numbers
much more concisely.
1. Scientific notation
In scientific notation each number is written in the form
a Ã— 10
where a is a number between 1 and 10 and n is a positive or
negative whole number. Some numbers in scientific notation are
5Ã—10, 2.67Ã—10,
7.90Ã—10
To understand scientific notation you need to be aware that 10
= 10, 10 = 100, 10
= 1000, 10 = 10000 and so on, and also that
and so on.
You also need to remember how simple it is to multiply a
number by powers of 10. For example to multiply 3.45 by 10, the
decimal point is moved one place to the right to give 34.5. To
multiply 29.65 by 100, the decimal point is moved two places to
the right to give 2965. In general, to multiply a number by 10
the decimal place is moved n places to the right if n is a
positive whole number and n places to the left if n is a negative
whole number. It may be necessary to insert additional zeros to
make up the required number of digits.
Example
The following numbers are given in scientific notation. Write
them out fully.
a) 5Ã—10
b) 2.67Ã—10
c) 7.90Ã—10
Solution
a) 5Ã—10 = 5Ã—1000 = 5000.
b) 2.67Ã—10 = 26700
c) 7.90Ã—10 = 0.00790
Example
Express each of the following numbers in scientific notation.
a) 5670000
b) 0.0098
Solution
a) 5.67Ã—10
b) 9.8Ã—10
Exercises
1. Express each of the following in scientific notation.
a) 0.00254
b) 82
c) 0.342
d) 1000000
Answers
1. a) 2.54Ã—10, b) 8.2Ã—10, c) 3.42Ã—10, d) 1Ã—10
or simply 10
2. Using a calculator
Students often have difficulty using a calculator to deal with
scientific notation. You may need to refer to your calculator
manual to ensure that you are entering numbers correctly. You
should also be aware that your calculator can display a number in
lots of different forms including scientific notation. Usually a
MODE button is used to select the appropriate format. Commonly
the EXP button is used to enter numbers in scientific notation.
(EXP stands for exponent which is another name for a power). A
number like 3.45Ã—10
is entered as 3.45 EXP 7 and might appear in the calculator
window as 3.45. Alternatively your calculator may
require you to enter the number as 3.45E7 and it may be displayed
in the same way. You should seek help if in doubt.
Computer programming languages use similar notation. For
example
8.25 Ã—10 may be programmed as 8.25E7
and
9.1Ã—10 may be programmed as 9.1E3
Again, you need to take care and check the required syntax
carefully.
A common error is to enter incorrectly numbers which are
simply powers of 10. For example, the number 10
is erroneously entered as 10E7 which means 10Ã—10,
that is 10. The number 10,
meaning 1Ã—10, should be entered as 1E7.
Check that you are using your calculator correctly by
verifying that
(3Ã—10)Ã—(2.76Ã—10)Ã—(10) = 8.28Ã—10
