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

Lesson Aims By the end of this lesson: You will be able to: Describe what an integer is Represent positive integers in binary using up to 8 Bits State three advantages of the binary number system Represent negative numbers using binary (Nat 5) Represent real numbers using binary (Nat 5)

The Decimal & Binary Systems Nat 4/5 The Decimal & Binary Systems We use the decimal(denary) or base 10 system This means we have 10 digits : 0-9 Computers use the binary or base 2 system There are only two digits: 0 and 1 Each figure is known as a bit Binary digit Because of our ten fingers? Off and On

Nat 4/5 Decimal Numbers Lets look at how a decimal number is made up: 173 Technically this is: Hundreds Tens Units 1 7 3 102 101 100 Hundreds Tens Units 1 7 3

Nat 4/5 Binary Numbers Because Binary is only made up of two digits (0 and 1) instead of 10 (as in Decimal), the column headings will be different. This is represented as: 24 23 22 21 20 16 8 4 2 1

Nat 4/5 Binary Numbers Lets look at how 173 is stored in binary: 1010 1101 128 64 32 16 8 4 2 1 The binary number is calculated by adding together the value of the columns with a 1. = 128 + 32 + 8 + 4 + 1 = 173

How to convert from Binary Nat 4/5 How to convert from Binary Create your table with the values in the top Insert your binary value into the table Add the place values that have a binary 1 in them: 64+32+4+2 = 102 128 64 32 16 8 4 2 1 1

Advantages of the Binary system Nat 4/5 Advantages of the Binary system There are less rules of arithmetic, this makes it easier to program. 0’s and 1’s are easier to represent digitally in computer systems. Any drop in voltage does not affect the data.

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. Advantages of Binary: There are less rules of arithmetic. 0’s and 1’s are easier to represent. Any drop in voltage doesn’t effect data.

What about negative numbers? Nat 5 What about negative numbers? Integers include:- -123, -89, 0, 45, 109876, etc Positive and negative integers are represented using two’s complement

Two’s complement Write -7 :- +7 using 8 bits 0000 0111 Nat 5 Two’s complement Write -7 :- +7 using 8 bits 0000 0111 Change 1’s and 0’s 1111 1000 Add 1 + 1 So -7 is 1111 1001

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

Real numbers Decimal Fractions:- 1/10 1/100 1/1000 …. As a fraction Nat 5 Real numbers Decimal Fractions:- 1/10 1/100 1/1000 …. As a fraction 0.1 0.01 0.001 …. As a decimal Binary Fractions:- ½ ¼ 1/8 0.5 0.25 0.125 …. as a fraction 0.1 0.01 0.001 …. as a decimal

Real numbers task Question 3 on Worksheet 1a

Floating point representation From Maths we know that:- 20.125 = .20125 x 102 Mantissa x Base Exponent In binary :- 20.125 = 10100.001 = .10100001 x 2 101 5 in decimal is 101 in binary

Floating point representation Nat 5 Floating point representation The computer stores the mantissa & exponent of a real number .10100001 x 2 101 Mantissa = 10100001 Exponent = 101

Nat 5 Floating point task Question 4 Worksheet 1a