Transcript Number Systems

Number Systems
Revision of conversations
What is a register
Addition
Complementation
Revision of conversions
Decimal
Binary
101012
Hexadecimal
34010
1011102
100111112
3816
45010
C6716
B2116
4410
Answers
Decimal
2110
34010
4610
15910
5610
45010
317510
284910
4410
Binary
101012
1010101002
1011102
100111112
001110002
1110000102
1100011001112
1011001000012
1011002
Hexadecimal
1516
15416
2E16
9F16
3816
1C216
C6716
B2116
2C16
Home Work

Convert the following
Decimal
Binary
1110112
Hexadecimal
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 Work
Find the following for registers A, B and C
1. 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. 10
B. 8
C. 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 +1=
1+0 =
1+1=
1+1+1
0
1
1
10
11
0
1
1
2
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
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
Decimal
10910
22910
6410
8910
Binary
One’s
Complement
Answers
Decimal
Binary
10910
11011012
One’s
Complement
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
-1010
-4210
-5510
-6010
Binary
Two’s
Complement
Answers
Decimal
Binary
-1010
000010102
Two’s
Complement
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
5610
8910
6710
2110
4910
Two’s
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;
11111111+
10101010

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;
1101111+
1101101
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
1. A right shift
2. 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