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Number Systems
Revision of conversationsWhat is a registerAdditionComplementation
Revision of conversionsDecimal Binary Hexadecim
al 101012
34010
1011102
100111112
3816
45010
C6716
B2116
4410
AnswersDecimal Binary Hexadecim
al 2110 101012 1516
34010 1010101002 15416
4610 1011102 2E16
15910 100111112 9F16
5610 001110002 3816
45010 1110000102 1C216
317510 1100011001112
C6716
284910 1011001000012
B2116
4410 1011002 2C16
Home WorkConvert the following
Decimal Binary Hexadecimal
1110112
5510
0000112
010101002
A9E16
19910
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 WorkFind the following for registers A,
B and C1. How many bits would each one
hold?2. How many combinations could
be made in each register?3. What would the range of
numbers be for each one?A. 10B. 8C. 14
Addition We could perform addition on
binary numbers
Here are some examples;
0+ 0+ 1+ 1+ 1+
0 1 0 1 1 1
0 1 1 10 11
REMEMBER If we remember the following
rules we will have no problems when performing additions
0 + 0 =
0
0 +1= 11 + 0 =
1
1 + 1 =
10
1 + 1 + 1
11
0112 3
Trying out additions
100+ 1001+ 111+
011 0011 001
Answers
100+ 1001+ 111+
011 0011 001
111 1100 1100
Home work
1110+ 0101+ 1101+
1111 0110 0100
0101+ 1000+ 1110+
0111 1001 1010
ComplementationComplementation 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
Decimal Binary One’s Complement
10910
22910
6410
8910
Answers
Decimal Binary One’s Complement
10910 11011012 00100102
22910 111001012 000110102
6410 10000002 01111112
8910 010110012 101001102
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,
Decimal Binary Two’s complement
4510 001011012 110100112
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 ;Decimal Binary Two’s
Complement
-1010
-4210
-5510
-6010
Answers
Decimal Binary Two’s Complement
-1010 000010102 111101102
-4210 001010102 110101102
-5510 001101112 110010012
-6010 001111002 110001002
Home Work Convert the following negative
numbers into binary using one’s and twos complement;
Decimal
Binary
One’s Complement
Two’s Complement
5610
8910
6710
2110
4910
Numerical OverflowAn 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?
11111111+
10101010
Numerical OverflowWhen 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 =
1101111+
1101101
What is Bit Shifting?Bit shifting is the process of moving
all the bits in a binary number
We have two shifts1. A right shift2. 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.
Left Shift
Right Shift
Ranges