﻿ Unit 1 Section 1 : Binary Numbers

Unit 1 Section 1 : Binary Numbers

We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers.

In base 10 we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

In base 2 we use only the digits 0 and 1.

Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents no current flowing whereas 1 represents a current flowing.

In base 10 we use a system of place values as shown below:

 1000 100 10 1 4 2 1 5 4 × 1000 + 2 × 100 + 1 × 10 + 5 × 1 3 1 0 2 3 × 1000 + 1 × 100 + 2 × 1

Note that, to obtain the place value for the next digit to the left, we multiply by 10.
If we were to add another digit to the front (left) of the numbers above, that number would represent 10 000s.

In base 2 we use a system of place values as shown below:

 64 32 16 8 4 2 1 1 0 0 0 0 0 0 1 × 64 = 64 1 0 0 1 0 0 1 1 × 64 + 1 × 8 + 1 × 1 = 73

Note that the place values begin with 1 and are multiplied by 2 as you move to the left.

Once you know how the place value system works, you can convert binary numbers to base 10, and vice versa.

Example 1

Convert the following binary numbers to base 10:

 (a) 111 (b) 101 (c) 1100110

Example 2

Convert the following base 10 numbers into binary numbers:

 (a) 3 (b) 11 (c) 140

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.
Question 1
Convert the following binary numbers to base 10:
 (a) 110 (b) 1111 (c) 1001 (d) 1101 (e) 10001 (f) 11011 (g) 1111111 (h) 1110001 (i) 10101010 (j) 11001101 (k) 111000111 (l) 1100110
Question 2
Convert the following base 10 numbers to binary numbers:
 (a) 9 (b) 8 (c) 14 (d) 17 (e) 18 (f) 30 (g) 47 (h) 52 (i) 67 (j) 84 (k) 200 (l) 500
Question 3
Convert the following base 10 numbers to binary numbers:
 (a) 5 (b) 9 (c) 17 (d) 33
 What will be the next base 10 number that will fit this pattern?
Question 4
Convert the following base 10 numbers to binary numbers:
 (a) 3 (b) 7 (c) 15 (d) 31
 What is the next base 10 number that will continue your binary pattern?
Question 5
A particular binary number has 3 digits.
(a) What are the largest and smallest possible binary numbers?
 largest: smallest:
(b) Convert these numbers to base 10.
Question 6
When a particular base 10 number is converted it gives a 4-digit binary number. What could the original base 10 number be?
Write the numbers as a list. e.g. 21, 22, 23, etc
Question 7
A 4-digit binary number has 2 zeros and 2 ones.
List the numbers separated by commas. e.g. 21, 22, 28, 30
 (a) List all the possible binary numbers with these digits. (b) Convert these numbers to base 10.
Question 8
A binary number has 8 digits and is to be converted to base 10.
 (a) What is the largest possible base 10 answer? (b) What is the smallest possible base 10 answer?
Question 9
The base 10 number 999 is to be converted to binary. How many more digits does the binary number have than the number in base 10?
Question 10
Calculate the difference between the base 10 number 11111 and the binary number 11111, giving your answer in base 10.