Download 6 - Number Theory 2

Number Systems - Part II
CS 215 Lecture # 6
Conversion from Decimal
Let N be the number to be converted.
We can write N as
N = q*b + r
where
q = quotient
b = the base into which N is to be
converted
r = remainder
Example 6.1
Convert 8010 to binary (base 2)
80 = 40*2 + 0
40 = 20*2 + 0
20 = 10*2 + 0
10 = 5*2 + 0
5 = 2*2 + 1
2 = 1*2 + 0
1 = 0*2 + 1
Answer: 10100002
Example 6.2
Convert 8010
80 = 10*8 +
10 = 1*8 +
1 = 0*8 +
Answer: 8010
to octal(base 8)
0
2
1
= 1208
Also, from binary:
8010 = 10100002 = 1 010 000 = 120
Example 6.3
Convert 8010
80 = 5*16 +
5 = 0*16 +
Answer: 8010
to hexadecimal(base 16)
0
5
= 5016
Also, from binary:
8010 = 10100002 = 101 0000 = 50
Conversion from Decimal
Decimal  Binary or Hexadecimal
Repeated division by 2 or 16
Example: Convert 292 to
hexadecimal.
292/16 = 18 R 4
18/16 = 1 R 2
1/16 = 0 R 1
292 = 12416
When the quotient is less
than 16, the process ends
Conversion from Decimal

Convert 42 to hexadecimal



42/16 = 2 R 10 (but 10 = a16)
2/16 = 0 R 2
42 = 2a16
Convert 109 to binary







109/2 = 54 R 1
54/2 = 27 R 0
27/2 = 13 R 1
13/2 = 6 R 1
6/2 = 3 R 0
3/2 = 1 R 1
1/2 = 0 R 1
109 = 11011012
Shortcut: convert to hexadecimal

It is sometimes easier to convert
from binary to hexadecimal, then to
decimal, instead of converting
directly to decimal
11101010110110
= 0011 1010 1011 0110
= 3
a
b
6
= 3*16^3 + 10*16^2 + 11*16 + 6
= 3*4096 + 10*256 + 176 + 6
= 12198 + 2560 + 182
= 12198 + 2742 = 14940
Shortcut: convert to hexadecimal

The converse is also true:
converting from decimal to
hexadecimal, then to binary
generally requires far fewer steps
than converting directly to binary
278/16 = 17 R 6
17/16 = 1 R 1
1/16 = 0 R 1
278 = 11616 = 0001 0001 01102 =
1000101102
Example 6.4
Convert the following:
5010 = ____16
5010 = ____8
5010 = ____2
5010 = ____5
Conversion of Fractions

A decimal improper fraction a/b is
converted into a different base b by

converting it into a mixed fraction




a/b = d e/f
e.g. 13/8 = 1 5/8
convert d into the required base using the
techniques we have already discussed
convert e/f (now a proper fraction) into the
required base by following the steps below
the answer is in the format ...f1f0.f-1f-2...
Example 6.5
Convert 2 5/8 to binary
2*(5/8) = 1 + 1/4
2*(1/4) = 0 + 1/2
2*(1/2) = 1 + 0
Answer: 10.101
Example 6.6
Convert 3.6 to binary
2*.6 = 1 + .2
2*.2 = 0 + .4
2*.4 = 0 + .8
Answer: 11.10011001...
2*.8 = 1 + .6
2*.6 = 1 + .2
2*.2 = 0 + .4
2*.4 = 0 + .8
2*.8 = 1 + .6
Converting back into decimal
10.101
fraction part
= 1*(1/2)1 + 0*(1/2)2 + 1*(1/2)3
= .625 or 5/8
integer part
= 1*(21) + 0*(20)
= 2
Answer: 2 5/8
Scientific Notation
Given a number N, we can express N
as
N = s*rx
where
s = mantissa or significand
r = radix
x = exponent


The mantissa can be made an
integer by multiplying it with a fixed
power of the radix and subtracting
the appropriate integer constant
from the exponent
For example,
2.358 * 101
= 2358 * 10-2
Normalization

Standard scientific notation defines
the normal form to meet the
following rule.
1  s < r
where s = significand
r = radix

A representation is in normal form if
the radix point occurs just to the
right of the first significant symbol
Example 6.7
2358 * 10-2
= 2.358 * 101
1525 * 10-1
= 1.525 * 102
Representing Numbers



Choosing an appropriate representation is
a critical decision a computer designer has
to make
The chosen representation must allow for
efficient execution of primitive operations
For general-purpose computers, the
representation must allow efficient
algorithms for


addition of two integers
determination of additive inverse
Representing Numbers





With a sequence of N bits, there are
2N unique representations
Each memory cell can hold N bits
The size of the memory cell
determines the number of unique
values that can be represented
The cost of performing operations
also increases as the size of the
memory cell increases
It is reasonable to select a memory
cell size such that numbers that are
frequently used are represented
Binary Representation


The binary, weighted positional
notation is the natural way to
represent non-negative numbers.
MAL numbers the bits from right to
left, beginning with 0 as the least
significant digit.
31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
Modulo Arithmetic



Consider the set of
number {0, … ,7}
Suppose all arithmetic
operations were finished
by taking the result
modulo 8
3 + 6 = 9, 9 mod 8 = 1


7
1
6
2
3+6=1
3*5 = 15, 15 mod 8 = 7

0
3*5=7
5
3
4
Modulo Arithmetic: Additive Inverse




What is the additive
inverse of 7?
7+x=0
7+1=0
0 and 4 are their own
additive inverses
0
7
1
6
2
5
3
4