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

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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)

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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 Bi nary digi t Nat 4/5 Off and On Because of our ten fingers?

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Decimal Numbers Lets look at how a decimal number is made up: 173 Technically this is: Nat 4/5 HundredsTensUnits 173 10 2 10 1 10 0 HundredsTensUnits 173

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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: Nat 4/5 2424 23232 2121 2020 168421

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Binary Numbers Lets look at how 173 is stored in binary: 1010 1101 Nat 4/5 1286432168421 10101101 The binary number is calculated by adding together the value of the columns with a 1. = 128 + 32 + 8 + 4 + 1 = 173

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How to convert from Binary 1. Create your table with the values in the top 2. Insert your binary value into the table 3. Add the place values that have a binary 1 in them: 64+32+4+2 = 102 Nat 4/5 1286432168421 01100110

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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. Nat 4/5

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Advantages of Binary: 1.There are less rules of arithmetic. 2.0’s and 1’s are easier to represent. 3.Any drop in voltage doesn’t effect data. 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. Nat 4/5

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

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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

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

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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 Nat 5

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Real numbers task Question 3 on Worksheet 1a

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Floating point representation From Maths we know that:- 20.125 =.20125 x 10 2 Mantissa x Base Exponent In binary :- 20.125 = 10100.001 =.10100001 x 2 101 5 in decimal is 101 in binary

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

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Floating point task Question 4 Worksheet 1a Nat 5

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