The document discusses different number systems including binary, decimal, hexadecimal, and octal. It covers the basics of each system including their radix value and digit sets. The key topics covered include: 1) Converting between different number systems such as decimal to binary and vice versa. 2) Performing arithmetic operations like addition, subtraction, multiplication in binary. This includes using binary complements for subtraction. 3) Representing signed binary numbers and arithmetic on signed binary numbers.

1. NumberSystemsMuhaiminBinMunir
DeptofCSE,MIST

2. 1.
Counting

3. 3
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1

4. 2.a
NumberSystems
Decimal Numbers

5. Basics-Base10NumberSystem
5
Consists of TWOThings:
✘ A BASE or RADIXValue
✘ A SET of DIGITS
✗ Digits are symbols representing allvalues
less than the radix value.
Example is the Common DecimalSystem:
✘ RADIX (BASE) =10
✘ Digit Set ={0,1,2,3,4,5,6,7, 8,9}

6. 231. 54
6
hundreds | tens | ones tenth | hundredth
2x100+3x10+1x1+5/10+4/100
2x10²+3x10¹+1x100+ 5x10-1+4x10-2
200 + 30 + 1 + 0.5 + 0.04

7. GeneralRepresentation
7
an - 1rn- 1+an - 2rn - 2 +an - 3rn - 3+….+a2r2 +a1r1+a0r0
+
a-1r-1+a-2r-2 +a-3r-3 +…..+a-mr-m
Here,
r =radix or base
ai =digit in i-thposition/coefficients
n=number of digits before decimalplace

8. 2.b
NumberSystems
Binary Numbers

9. Basics-Base2NumberSystem
✘ RADIX (BASE) =2
✘ Digit Set ={0,1}
Convert a Binary Number into a Decimal Number
Just put the radix value two (r=2)in thegeneral
representation of decimalnumbers.
9

10. 1011. 01
8’s | 4’s | 2’s | 1’s ½ | ¼th
1x8+0x4+1x2+1x1+ 0/2+1/4
1x23+0x22+1x21+1x20+ 0x2-1 +1x2-2
0 + 0.258 + 0 + 2 + 1 +
1
20
=11. 5

11. 2.c
NumberSystems
Hexadecimal Numbers &
Octal Numbers

12. Basics
12
Octal Numbers(Base 8 Number System)
✘ RADIX (BASE) =8
✘ Digit Set ={0,1,2,3,4,5,6, 7}
Hexadecimal Numbers(Base 16Number System)
✘ RADIX (BASE) =16
✘ Digit Set ={0,1,2,3,4,5,6,7,8,9,A,B, C,D,E,F}

13. HexadecimalandOctaltoDecimalNumberConversion
Hexadecimal Octal
6 E 7 1 5 . 2
256’s|16’s|1’s 8’s|1’s. ⅛th
6x256+14(E)x16+7x1 1x8+5x1+2/8
=1536+224+7 =8+5+0.25
=1767 =13.25
13

14. 3
Conversions

15. ✘ Divide the number
by the ‘Base’(2)
✘ Take the remainder
(either 0 or 1)as a
coefficient
✘ Take the quotient
and repeat the
division
Decimal(Integer)toBinary
Example: 14
Quo Rem Coeff
14/2 7 0 a0 =0
7/2 3 1 a1=1
3/2 1 1 a2 =1
1/2 0 1 a3 =1
15
(14)10=(a3a2a1a0)2 =(1110)2

16. ✘ Multiply the number
by the ‘Base’(2)
✘ Take the integer
(either 0 or 1)asa
coefficient
✘ Take the resultant
fraction and repeat
the multiplication
Decimal(Fraction)toBinary
Coeff
16
a-1=1
a-2 =0
-3
Example: 0.625
Int Frac
0.625 x 2 1 .25
0.25 x 2 0 .5
0.5 x 2 1 .0 a =1
(0.625)10=(0.a-1a-2a-3)2 =(0.101)2

17. DecimaltoOctal
Coeff
a-1=2
Example: 0.3125
Int Frac
0.3125 x 8 2 .5
0.5 x 8 4 .0 a-2 =4
(0.3125)10=(0.a-1a-2)8=(0.24)8
Coeff
17
175/8
21/8
Example: 175
Quo Rem
21 7 a0 =7
2 5 a1 =5
2/8 0 2 a2 =2
(175)10=(a2a1a0)8 =(275)8

18. ✘ Represent the binary
number in groups of 3
bits.
✗ 8 =23Hence, 3 bits
✘ Replace 3bit
representation with
octal digits.
BinarytoOctalandVice-versa
Example: (10110.01)2
010 110. 010
2 6 . 2
0 1 2 3 4 5 6 7
000 001 010 011 100 101 110 111
18

19. ✘ Convert octal to
binary as an
intermediate step.
✘ Convert binary to
hexadecimal.
OctaltoHexadecimalandVice-versa
Example: (26.2)8
010 110. 010
00010110. 0100
1 6 . 4
19

20. “Practiceputsbrainsinyourmuscles”
20
SamSnead

21. HowtoCountto1000bytwohands

22. 4.a
BinaryArithmetic
Addition and Multiplication

23. +
111111
111101 61
10111 +23
1010100 84

24. 101
11
5
3
101 15
+ 1010
1111
x x

25. 4.b
BinaryArithmetic
Subtraction and Complements

26. SubtractionusingComplements
Two types (for each base-rsystem)
✘ Diminishing radix complement (r - 1’scomplement)
✘ Radix complement (r’scomplement)
For n-digit numberN
rn- 1- N
rn- N
r - 1’scomplement
r’s complement
26

27. DiminishedRadixComplement
27
✘ Example for 6-digit decimal numbers:
9’s complement is (rn –1)–N =(106–1)–N =999999 –N
9’s complement of 546700:
999999 –546700 =453299
✘ Example for 7-digit binary numbers:
1’scomplement is (rn –1)–N =(27–1)–N =1111111–N
1’scomplement of 1011000:
1111111–1011000=0100111

28. RadixComplement
28
The r's complement ofan n-digit number Nin base r is
defined as rn–Nfor N≠0and as 0for N=0.
The r's complement can also be obtained by adding 1to
the (r −1)'s complement, since rn–N=[(rn−1)–N]+1
Example: 10’scomplement of 246700is
106- N =1000000 - 246700 =753300

29. ✘ Take 1’scomplement then add 1
✘ Toggle all bits to the left of the first ‘1’from the
right
RadixComplement(BinaryNumbers)
1
01010000 29
10110000 10110010
01001111
01001110

30. M–N
✘ Add Mto r’s complement of N
sum=M+(rn–N)=M–N +rn
✘ If M>N,Sum will have an end carry rn,discard it
✘ If M<N,Sum will not have an end carry and
sum=rn–(N–M)(r’scomplementof N–M)
So M–N=–(r’scomplementof Sum)
SubtractionwithradixComplement

31. 65438-5623
65438
10’scomplement
5623
94377
15981531

32. 05623
5623-65438
32
10’scomplement
65438
34562
40185
Therefore,theansweris:–(10'scomplementof40185)=−59815.

33. 10110010-10011111
10110010
2’scomplement
10011111
01100001
10001001133

34. 10011111
10011111-10110010
Therefore,theanswer:–(2'scomplementof11101101)=−10011
34
2’scomplement
10110010
01001110
11101101

35. Subtraction of unsigned numbers can also be done by
means of the (r−1)'scomplement.
Remember that the (r−1)'scomplement is oneless
then the r 'scomplement.
SubtractionwithDiminishedradixComplement

36. 36
10110010
10110010-10011111
1’scomplement
10011111
01100000
100010010
00010010+1=10011

37. 37
10011111
10011111-10110010
1’scomplement
10110010
01001101
11101100
Therefore,theanswer:–(1'scomplementof11101100)=−10011

38. 4.c
BinaryArithmetic
Signed Binary Numbers

39. To represent negative integers, we need a notation for
negative values.
It is customary to represent the sign with a bit placed
in the leftmost position of the number since binary
digits.
The convention is to make the sign bit 0 for positive
and 1for negative.
SignedBinaryNumbers

40. SignedBinaryNumbersArithmeticAddiTion
40
-6 11111010
+5 00000101 - 5 11111011
+11 00001011 +11 00001011
+16 00010000 +6100000110
+5 00000101 -5 11111011
-11 11110101 -11 11110101
-16111110000

41. SignedBinaryNumbersArithmeticSubtraction
41
Funfact: Signed Binary Subtraction is Basically
Binary Additions.
Cuz, (±A) - (+B) =(±A) +(-B)
And (±A) - (- B) =(±A) +(+ B)
(-6)-(-13) =11111010-11110011
=11111010+00001101
=00000111(+7)

42. Binary Code: ASCII
42
Character code
ASCII Chart
SelfStudyMaterials
Binary Code: BCD
Code
BCD Arithmetics,
Examples, Decimal
Arithmetic
Binary Code: Other
Decimal Codes
BCD (8421),2421,Excess 3,
8-4-2-1codings, Gray
Code

43. 43

44. THANKS!
Any questions?
You can find me atmuhaiminbinmunir@gmail.com
44