1. Nat 4/5 Computing Science Lesson 1: Binary
Computer Systems
Nat 4/5 Computing Science
Lesson 1:
Binary

2. Lesson Aims By the end of this lesson: You will be able to:
Use binary to represent and store positive integers
Represent positive integers in binary using up to 8 Bits
Represent negative numbers using binary (Nat 5)
Represent real numbers using binary (Nat 5)

3. The two-state machine
Computers and storage devices (hard drives, USB Flash drives, CD ROMs, DVDs) have components that have only two states:-
On and off
1 represents on
0 represents off

4. The Decimal & Binary Systems
Nat 4/5
The Decimal & Binary Systems
We use the decimal or base 10 system
This means we have 10 digits : 0-9
Computers use the binary or base 2 system
There are two digits: 0 and 1
Each figure is known as a bit
Binary digit

5. Nat 4/5
Decimal Numbers
Lets look at how a decimal number is made up: 173
Hundreds
Tens
Units 1 7 3

6. Binary Numbers The column headings for binary are :- 128 64 32 16 8 4
Nat 4/5
Binary Numbers
The column headings for binary are :-
128 64 32 16 8 4 2 1

7. Converting Binary to decimal
Nat 4/5
Converting Binary to decimal
Lets look at how 173 is stored in binary:
128 64 32 16 8 4 2 1 =
= 173

8. Steps to convert from Binary to Decimal
Nat 4/5
Steps to convert from Binary to Decimal
Write the binary column headings
128 64 32 16 8 4 2 1 1
Write the binary number below the headings
Add the column headings that have a 1: = 102

9. Converting Decimal to binary
Nat 4/5
Converting Decimal to binary
Lets look at how to write the decimal value 145 in binary:-
128 64 32 16 8 4 2 1 =
= 145

10. Steps to convert from Decimal to Binary
Nat 4/5
Steps to convert from Decimal to Binary
Write the binary column headings
128 64 32 16 8 4 2 1 1
Write a 1 below each column heading that you want to use – write a 0 below the others
Check that the column with a 1 add to make your decimal number: = 146

11. Summary Binary only has two values– 0 & 1
Nat 4/5
Summary
Binary only has two values– 0 & 1
A single 0 or 1 is known as a bit
Binary Digit
The place values in binary start on the right at 1 and double every time going to the left.
Complete Worksheet 1

12. What about negative numbers?
Nat 5
What about negative numbers?
Integers include:-
-123, -89, 0, 45, , etc
Positive and negative integers are represented using a system called two’s complement
0 at the start = positive number
1 at the start = a negative number

13. Two’s complement Write -7 :- +7 using 8 bits 0000 0111
Nat 5
Two’s complement
Sign – or +
Write -7 :-
+7 using 8 bits
Change 1’s and 0’s
Add
So -7 is

14. Nat 5
Twos complement tasks
Questions 1 and 2 on Worksheet 1a

15. Real numbers Decimal Fractions:- 1/10 1/100 1/1000 …. fraction
Nat 5
Real numbers
Decimal Fractions:-
1/10 1/100 1/ …. fraction
…. decimal
Binary Fractions:-
½ ¼ / …. fraction
…. decimal
…. binary

16. Real numbers Write 10.5 in binary 8 4 2 1 . 0.5 0.25 0.125
Nat 5
Real numbers
Write 10.5 in binary
What is this number?
=7.75

17. Nat 5
Real numbers task
Question 3 on Worksheet 1a

18. Floating point representation
From Maths we know that:-
= x 102
Mantissa x Base Exponent
In binary :-
= = x 2 101
5 in decimal is 101 in binary

19. Floating point representation
Nat 5
Floating point representation
The computer stores the mantissa & exponent of a real number
x 2 101
Mantissa =
Exponent = 101

20. Nat 5
Floating point task
Question 4 Worksheet 1a