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 × 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

5×10, 2.67×10, 7.90×10

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) 5×10

b) 2.67×10

c) 7.90×10

Solution

a) 5×10 = 5×1000 = 5000.

b) 2.67×10 = 26700

c) 7.90×10 = 0.00790

Example

Express each of the following numbers in scientific notation.

a) 5670000

b) 0.0098

Solution

a) 5.67×10

b) 9.8×10

Exercises

1. Express each of the following in scientific notation.

a) 0.00254

b) 82

c) -0.342

d) 1000000

Answers

1. a) 2.54×10, b) 8.2×10, c) -3.42×10, d) 1×10 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 3.45×10 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 ×10 may be programmed as 8.25E7

and

9.1×10 may be programmed as 9.1E-3

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 10×10, that is 10. The number 10, meaning 1×10, should be entered as 1E7.

Check that you are using your calculator correctly by verifying that

(3×10)×(2.76×10)×(10) = 8.28×10