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Binary and all that…

CP4 Revision

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• Binary is a Base 2 Number System

• Only uses digits 0 and 1

• Each digit is called a BIT – short for Binary digIT

The Binary Number System

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Hexadecimal

• Shorthand way of writing binary numbers

• Base 16 number system

• Uses digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

• Each hex digit represents a 4-bit binary number.

• 6 = 0110

• C = 1100

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Hex Exercise

1. Convert 011010100011111016 into hex

2. Convert B216 into binary

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Characters

• Each character has a unique binary code.

• Most commonly used system is ASCII

• A = 01000001

• B = 01000010

• C = 01000011

• etc

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Positive Integers

• Calculate using the headings…

128 64 32 16 8 4 2 1

Example : Binary representation of 7410

128 64 32 16 8 4 2 1

0 1 0 0 1 0 1 0

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Negative Integers

• Two methods :– Sign and Magnitude– Two’s Complement

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Sign and Magnitude

• First bit is a Sign bit (0 for Positive; 1 for Negative).

• Remaining bits are the Size of the integer.

Sign 64 32 16 8 4 2 1

1 0 0 1 0 1 0 1

Example : -21

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Two’s Complement

• Three steps…– Treat as positive– Change all bits– Add 1

128 64 32 16 8 4 2 1

0 0 0 1 0 1 1 0

Example : -22

Step 1 :

Step 2:128 64 32 16 8 4 2 1

1 1 1 0 1 0 0 1

Step 3:

+1

1 1 1 0 1 0 1 0

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Real Numbers

• Two methods –– Fixed Point – Floating Point

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Fixed Point

• The position of the binary point remains fixed.• Eg.: 4 points for the Integer part and 4 bits for

the fraction…

S 4 2 1 1/2 1/4 1/8 1/16

0 1 0 1 . 0 1 0 0

= 5.25

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Floating Point

• Floating Point Format –

Mantissa x 2 Exponent

Mantissa is a Signed Fraction (Fixed Point)

Exponent is a Signed Integer

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Convert a real to Floating Point Form

• Example : 9.75• Convert into fixed point binary number…

• Add 0 bits on the right until the correct number of bits for the mantissa : 01001.1100000 (assume 12 bit mantissa, 4 bit exponent for this)

• Move binary point to left until it is after the sign bit – count how many moves - because that is the exponent

• 0.10011100000 and the exponent is 4• Final answer : 010011100000 0100

CP4

Sign 8 4 2 1 . .5 .25

0 1 0 0 1 . 1 1

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Binary Exercises

1. Using 8-bit Two’s Complement system…how would these numbers be represented?

1. 4310

2. -4310

2. Using Sign and Magnitude system, how would the above numbers be represented?

3. Convert your answers to [1] into hexadecimal.

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Binary Exercises

• [CP4 2006](i) Convert the 12 bit binary number 101111010111 to hexadecimal. [1]  

• (ii) Why is hexadecimal often used as an alternative to binary? [1]

• [CP4 2005]• In a certain computer, two's complement is used to represent

negative integers, using 8 bits.  • (i) Show how the number -710 is represented. [1]   • (ii) Showing your working, demonstrate that 1310 is the result of the

binary addition of -710 to +2010. [2]  • (b)In another computer, a sign/magnitude approach is used to

represent integers using 8 bits. Explain what is meant by the term sign/magnitude, giving a clearly labelled example [2]  

• (c)In another computer, the character '1' is stored as 00110001. The character '2' is stored as the next higher binary code (00110010) and so on. How will the character '5' be stored? [1]