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Number Systems. Revision of conversations What is a register Addition Complementation. Revision of conversions. Answers. Home Work. Convert the following. What is a Register?. A small amount of very fast computer memory Speed up computer programs. Store most commonly used values,
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Number Systems Revision of conversations What is a register Addition Complementation
Home Work • Convert the following
What is a Register? • A small amount of very fast computer memory • Speed up computer programs. • Store most commonly used values, • Computers load information into the registers, and then load it back to the main memory (load-store architecture)
Registers • If we have a 5 bit register the maximum number of bits it can store is 5 • Since it can hold 5bits we could have 25 combinations (32) • The range of the register would be 0 – 25-1 = 0 - 31
Working with Registers • If we had a 12 bit register how many bits could it hold? • How many combinations would the register be able to hold? • What would the range of the register be?
Home Work • Find the following for registers A, B and C • How many bits would each one hold? • How many combinations could be made in each register? • What would the range of numbers be for each one? • 10 • 8 • 14
Addition • We could perform addition on binary numbers • Here are some examples;
REMEMBER • If we remember the following rules we will have no problems when performing additions
Complementation • Complementation is a used to represent positive and negative numbers. In binary • This system requires numbers to be represented by a fixed number of bits. • There are two forms of complementation, one’s complement and two’s complement.
Ones Complement • One’s complement is used to represent negative numbers • Lets say we have 4510 • When using 8 bits 4510 = 001011012 • If we change 4510 to -4510 The binary representation changes by converting 0s to 1s and 1s to 0s; 001011012 110100102. after ones complement
Examples • Change the following to negative binary numbers using one’s complement
Two’s Complement • Two’s complement allows us to perform subtractions with binary numbers • With two’s complement we start converting 1s to 0s and 0s to 1s after the first 1 • Lets take the previous example of -4510,
Another example • Lets say we had the number 1710 and we want to change it to a -17 in two’s complement • First we convert 1710 to binary using an 8 bit register = 000100012 • Starting after the first 1 we convert the bits = 111011112
Examples • Change the following negative numbers to binary using two’s complement and an 8 bit register ;
Home Work • Convert the following negative numbers into binary using one’s and twos complement;
Numerical Overflow • An overflow is when something doesn’t fit in a certain space • Numeric overflow is when the storage for a calculation is too small to hold the result • For example we have an 8 bits register, if we add two binary numbers and the result turns out to be 9 bits it would not fit in the register
Example • Let’s say we have an 8 bit register • Add the following; • Do we have an overflow?
Numerical Overflow • When we have a numeric overflow we will have an error in our calculation • When we have an overflow we would need to remove the extra bit at the start of the number • Lets say we had a 7 bit register and the result of a calculation is 11001100 the actual answer would be 1001100
Example • Let’s say we have a 7 bit register • Add the following; • Do we have an overflow? • Actual answer =
What is Bit Shifting? • Bit shifting is the process of moving all the bits in a binary number • We have two shifts • A right shift • A left shift • The right shift would divide the number while the left would multiply it
Right Shift • The right shift is used for • If we shift the byte 001101112 left once, we get 011011102. If we shift 101100112, right by three places, we get 000101102. Notice in the right-shift example that bits that are shifted out of the byte are lost. This also occurs with left-shifting: if any bits are shifted outside of the "boundaries" of the type in use (eight bits, for a byte), they are lost.
