Infographic showing computer number system conversions between Decimal, Binary, Octal, and Hexadecimal with binary arithmetic operations.
Infographic: Quick reference guide for computer number system conversions (Decimal, Binary, Octal, Hexadecimal) and binary arithmetic rules.

Computer Number System Conversion and Binary Arithmetic: Complete Solved Exercises

Master computer number systems with fully solved step-by-step calculation processes on Decimal, Binary, Octal, Hexadecimal conversions, and Binary Arithmetic (Addition, Subtraction, Multiplication, Division, and Simplification).

Part 1: Number System Conversions

(A) Decimal to Binary Conversion

To convert a decimal number to binary, repeatedly divide the number by 2 and write down the remainders from bottom to top.

i. (45)10

  • 45 ÷ 2 = 22, Remainder 1
  • 22 ÷ 2 = 11, Remainder 0
  • 11 ÷ 2 = 5, Remainder 1
  • 5 ÷ 2 = 2, Remainder 1
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(45)10 = (101101)2

ii. (88)10

  • 88 ÷ 2 = 44, Remainder 0
  • 44 ÷ 2 = 22, Remainder 0
  • 22 ÷ 2 = 11, Remainder 0
  • 11 ÷ 2 = 5, Remainder 1
  • 5 ÷ 2 = 2, Remainder 1
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(88)10 = (1011000)2

iii. (180)10

  • 180 ÷ 2 = 90, Remainder 0
  • 90 ÷ 2 = 45, Remainder 0
  • 45 ÷ 2 = 22, Remainder 1
  • 22 ÷ 2 = 11, Remainder 0
  • 11 ÷ 2 = 5, Remainder 1
  • 5 ÷ 2 = 2, Remainder 1
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(180)10 = (10110100)2

iv. (207)10

  • 207 ÷ 2 = 103, Remainder 1
  • 103 ÷ 2 = 51, Remainder 1
  • 51 ÷ 2 = 25, Remainder 1
  • 25 ÷ 2 = 12, Remainder 1
  • 12 ÷ 2 = 6, Remainder 0
  • 6 ÷ 2 = 3, Remainder 0
  • 3 ÷ 2 = 1, Remainder 1
  • 1 ÷ 2 = 0, Remainder 1
(207)10 = (11001111)2

v. (444)10

  • 444 ÷ 2 = 222, Remainder 0
  • 222 ÷ 2 = 111, Remainder 0
  • 111 ÷ 2 = 55, Remainder 1
  • 55 ÷ 2 = 27, Remainder 1
  • 27 ÷ 2 = 13, Remainder 1
  • 13 ÷ 2 = 6, Remainder 1
  • 6 ÷ 2 = 3, Remainder 0
  • 3 ÷ 2 = 1, Remainder 1
  • 1 ÷ 2 = 0, Remainder 1
(444)10 = (110111100)2

vi. (875)10

  • 875 ÷ 2 = 437, Remainder 1
  • 437 ÷ 2 = 218, Remainder 1
  • 218 ÷ 2 = 109, Remainder 0
  • 109 ÷ 2 = 54, Remainder 1
  • 54 ÷ 2 = 27, Remainder 0
  • 27 ÷ 2 = 13, Remainder 1
  • 13 ÷ 2 = 6, Remainder 1
  • 6 ÷ 2 = 3, Remainder 0
  • 3 ÷ 2 = 1, Remainder 1
  • 1 ÷ 2 = 0, Remainder 1
(875)10 = (1101101011)2

vii. (1406)10

  • 1406 ÷ 2 = 703, Remainder 0
  • 703 ÷ 2 = 351, Remainder 1
  • 351 ÷ 2 = 175, Remainder 1
  • 175 ÷ 2 = 87, Remainder 1
  • 87 ÷ 2 = 43, Remainder 1
  • 43 ÷ 2 = 21, Remainder 1
  • 21 ÷ 2 = 10, Remainder 1
  • 10 ÷ 2 = 5, Remainder 0
  • 5 ÷ 2 = 2, Remainder 1
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(1406)10 = (10101111110)2

viii. (920)10

  • 920 ÷ 2 = 460, Remainder 0
  • 460 ÷ 2 = 230, Remainder 0
  • 230 ÷ 2 = 115, Remainder 0
  • 115 ÷ 2 = 57, Remainder 1
  • 57 ÷ 2 = 28, Remainder 1
  • 28 ÷ 2 = 14, Remainder 0
  • 14 ÷ 2 = 7, Remainder 0
  • 7 ÷ 2 = 3, Remainder 1
  • 3 ÷ 2 = 1, Remainder 1
  • 1 ÷ 2 = 0, Remainder 1
(920)10 = (1110011000)2

ix. (2090)10

  • 2090 ÷ 2 = 1045, Remainder 0
  • 1045 ÷ 2 = 522, Remainder 1
  • 522 ÷ 2 = 261, Remainder 0
  • 261 ÷ 2 = 130, Remainder 1
  • 130 ÷ 2 = 65, Remainder 0
  • 65 ÷ 2 = 32, Remainder 1
  • 32 ÷ 2 = 16, Remainder 0
  • 16 ÷ 2 = 8, Remainder 0
  • 8 ÷ 2 = 4, Remainder 0
  • 4 ÷ 2 = 2, Remainder 0
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(2090)10 = (10000101010)2

x. (555)10

  • 555 ÷ 2 = 277, Remainder 1
  • 277 ÷ 2 = 138, Remainder 1
  • 138 ÷ 2 = 69, Remainder 0
  • 69 ÷ 2 = 34, Remainder 1
  • 34 ÷ 2 = 17, Remainder 0
  • 17 ÷ 2 = 8, Remainder 1
  • 8 ÷ 2 = 4, Remainder 0
  • 4 ÷ 2 = 2, Remainder 0
  • 2 ÷ 2 = 1, Remainder 0
  • 1 ÷ 2 = 0, Remainder 1
(555)10 = (1000101011)2

(B) Binary to Decimal Conversion

Multiply each binary digit by powers of 2 (20, 21, 22, ...) starting from the rightmost position and sum the results.

i. (1010)2

  • 1 × 23 = 8
  • 0 × 22 = 0
  • 1 × 21 = 2
  • 0 × 20 = 0
  • Sum = 8 + 0 + 2 + 0 = 10
(1010)2 = (10)10

ii. (11100)2

  • 1 × 24 = 16
  • 1 × 23 = 8
  • 1 × 22 = 4
  • 0 × 21 = 0
  • 0 × 20 = 0
  • Sum = 16 + 8 + 4 + 0 + 0 = 28
(11100)2 = (28)10

iii. (10011)2

  • 1 × 24 = 16
  • 0 × 23 = 0
  • 0 × 22 = 0
  • 1 × 21 = 2
  • 1 × 20 = 1
  • Sum = 16 + 0 + 0 + 2 + 1 = 19
(10011)2 = (19)10

iv. (101110)2

  • 1 × 25 = 32
  • 0 × 24 = 0
  • 1 × 23 = 8
  • 1 × 22 = 4
  • 1 × 21 = 2
  • 0 × 20 = 0
  • Sum = 32 + 0 + 8 + 4 + 2 + 0 = 46
(101110)2 = (46)10

v. (101011)2

  • 1 × 25 = 32
  • 0 × 24 = 0
  • 1 × 23 = 8
  • 0 × 22 = 0
  • 1 × 21 = 2
  • 1 × 20 = 1
  • Sum = 32 + 0 + 8 + 0 + 2 + 1 = 43
(101011)2 = (43)10

vi. (10110111)2

  • 1 × 27 = 128
  • 0 × 26 = 0
  • 1 × 25 = 32
  • 1 × 24 = 16
  • 0 × 23 = 0
  • 1 × 22 = 4
  • 1 × 21 = 2
  • 1 × 20 = 1
  • Sum = 128 + 0 + 32 + 16 + 0 + 4 + 2 + 1 = 183
(10110111)2 = (183)10

vii. (110011)2

  • 1 × 25 = 32
  • 1 × 24 = 16
  • 0 × 23 = 0
  • 0 × 22 = 0
  • 1 × 21 = 2
  • 1 × 20 = 1
  • Sum = 32 + 16 + 0 + 0 + 2 + 1 = 51
(110011)2 = (51)10

viii. (100001)2

  • 1 × 25 = 32
  • 0 × 24 = 0
  • 0 × 23 = 0
  • 0 × 22 = 0
  • 0 × 21 = 0
  • 1 × 20 = 1
  • Sum = 32 + 0 + 0 + 0 + 0 + 1 = 33
(100001)2 = (33)10

ix. (101010)2

  • 1 × 25 = 32
  • 0 × 24 = 0
  • 1 × 23 = 8
  • 0 × 22 = 0
  • 1 × 21 = 2
  • 0 × 20 = 0
  • Sum = 32 + 0 + 8 + 0 + 2 + 0 = 42
(101010)2 = (42)10

x. (111110)2

  • 1 × 25 = 32
  • 1 × 24 = 16
  • 1 × 23 = 8
  • 1 × 22 = 4
  • 1 × 21 = 2
  • 0 × 20 = 0
  • Sum = 32 + 16 + 8 + 4 + 2 + 0 = 62
(111110)2 = (62)10

Complete Number System Conversions: Decimal, Octal, Hexadecimal, and Binary Step-by-Step Calculations

Detailed step-by-step calculation processes for Decimal to Octal, Octal to Decimal, Decimal to Hexadecimal, Hexadecimal to Decimal, Binary to Octal/Hexadecimal, Octal to Binary/Hexadecimal, and Hexadecimal to Octal conversions.

(C) Decimal to Octal Conversion

To convert a decimal number to octal, repeatedly divide the number by 8 and record the remainders from bottom to top.

i. (56)10

  • 56 ÷ 8 = 7, Remainder 0
  • 7 ÷ 8 = 0, Remainder 7
(56)10 = (70)8

ii. (123)10

  • 123 ÷ 8 = 15, Remainder 3
  • 15 ÷ 8 = 1, Remainder 7
  • 1 ÷ 8 = 0, Remainder 1
(123)10 = (173)8

iii. (437)10

  • 437 ÷ 8 = 54, Remainder 5
  • 54 ÷ 8 = 6, Remainder 6
  • 6 ÷ 8 = 0, Remainder 6
(437)10 = (665)8

iv. (279)10

  • 279 ÷ 8 = 34, Remainder 7
  • 34 ÷ 8 = 4, Remainder 2
  • 4 ÷ 8 = 0, Remainder 4
(279)10 = (427)8

v. (555)10

  • 555 ÷ 8 = 69, Remainder 3
  • 69 ÷ 8 = 8, Remainder 5
  • 8 ÷ 8 = 1, Remainder 0
  • 1 ÷ 8 = 0, Remainder 1
(555)10 = (1053)8

vi. (684)10

  • 684 ÷ 8 = 85, Remainder 4
  • 85 ÷ 8 = 10, Remainder 5
  • 10 ÷ 8 = 1, Remainder 2
  • 1 ÷ 8 = 0, Remainder 1
(684)10 = (1254)8

vii. (808)10

  • 808 ÷ 8 = 101, Remainder 0
  • 101 ÷ 8 = 12, Remainder 5
  • 12 ÷ 8 = 1, Remainder 4
  • 1 ÷ 8 = 0, Remainder 1
(808)10 = (1450)8

viii. (901)10

  • 901 ÷ 8 = 112, Remainder 5
  • 112 ÷ 8 = 14, Remainder 0
  • 14 ÷ 8 = 1, Remainder 6
  • 1 ÷ 8 = 0, Remainder 1
(901)10 = (1605)8

ix. (1723)10

  • 1723 ÷ 8 = 215, Remainder 3
  • 215 ÷ 8 = 26, Remainder 7
  • 26 ÷ 8 = 3, Remainder 2
  • 3 ÷ 8 = 0, Remainder 3
(1723)10 = (3273)8

x. (2078)10

  • 2078 ÷ 8 = 259, Remainder 6
  • 259 ÷ 8 = 32, Remainder 3
  • 32 ÷ 8 = 4, Remainder 0
  • 4 ÷ 8 = 0, Remainder 4
(2078)10 = (4036)8

(D) Octal to Decimal Conversion

Multiply each octal digit by powers of 8 (80, 81, 82, ...) from right to left and sum the products.

i. (114)8

  • 1 × 82 = 64
  • 1 × 81 = 8
  • 4 × 80 = 4
  • Sum = 64 + 8 + 4 = 76
(114)8 = (76)10

ii. (234)8

  • 2 × 82 = 128
  • 3 × 81 = 24
  • 4 × 80 = 4
  • Sum = 128 + 24 + 4 = 156
(234)8 = (156)10

iii. (444)8

  • 4 × 82 = 256
  • 4 × 81 = 32
  • 4 × 80 = 4
  • Sum = 256 + 32 + 4 = 292
(444)8 = (292)10

iv. (567)8

  • 5 × 82 = 320
  • 6 × 81 = 48
  • 7 × 80 = 7
  • Sum = 320 + 48 + 7 = 375
(567)8 = (375)10

v. (601)8

  • 6 × 82 = 384
  • 0 × 81 = 0
  • 1 × 80 = 1
  • Sum = 384 + 0 + 1 = 385
(601)8 = (385)10

vi. (723)8

  • 7 × 82 = 448
  • 2 × 81 = 16
  • 3 × 80 = 3
  • Sum = 448 + 16 + 3 = 467
(723)8 = (467)10

vii. (1230)8

  • 1 × 83 = 512
  • 2 × 82 = 128
  • 3 × 81 = 24
  • 0 × 80 = 0
  • Sum = 512 + 128 + 24 + 0 = 664
(1230)8 = (664)10

viii. (2233)8

  • 2 × 83 = 1024
  • 2 × 82 = 128
  • 3 × 81 = 24
  • 3 × 80 = 3
  • Sum = 1024 + 128 + 24 + 3 = 1179
(2233)8 = (1179)10

ix. (732)8

  • 7 × 82 = 448
  • 3 × 81 = 24
  • 2 × 80 = 2
  • Sum = 448 + 24 + 2 = 474
(732)8 = (474)10

x. (711)8

  • 7 × 82 = 448
  • 1 × 81 = 8
  • 1 × 80 = 1
  • Sum = 448 + 8 + 1 = 457
(711)8 = (457)10

(E) Decimal to Hexadecimal Conversion

Repeatedly divide by 16. Note: 10=A, 11=B, 12=C, 13=D, 14=E, 15=F.

i. (76)10

  • 76 ÷ 16 = 4, Remainder 12 (C)
  • 4 ÷ 16 = 0, Remainder 4
(76)10 = (4C)16

ii. (106)10

  • 106 ÷ 16 = 6, Remainder 10 (A)
  • 6 ÷ 16 = 0, Remainder 6
(106)10 = (6A)16

iii. (321)10

  • 321 ÷ 16 = 20, Remainder 1
  • 20 ÷ 16 = 1, Remainder 4
  • 1 ÷ 16 = 0, Remainder 1
(321)10 = (141)16

iv. (808)10

  • 808 ÷ 16 = 50, Remainder 8
  • 50 ÷ 16 = 3, Remainder 2
  • 3 ÷ 16 = 0, Remainder 3
(808)10 = (328)16

v. (2047)10

  • 2047 ÷ 16 = 127, Remainder 15 (F)
  • 127 ÷ 16 = 7, Remainder 15 (F)
  • 7 ÷ 16 = 0, Remainder 7
(2047)10 = (7FF)16

vi. (2019)10

  • 2019 ÷ 16 = 126, Remainder 3
  • 126 ÷ 16 = 7, Remainder 14 (E)
  • 7 ÷ 16 = 0, Remainder 7
(2019)10 = (7E3)16

vii. (5544)10

  • 5544 ÷ 16 = 346, Remainder 8
  • 346 ÷ 16 = 21, Remainder 10 (A)
  • 21 ÷ 16 = 1, Remainder 5
  • 1 ÷ 16 = 0, Remainder 1
(5544)10 = (15A8)16

(F) Hexadecimal to Decimal Conversion

Multiply each hex digit by powers of 16 (160, 161, 162, ...) from right to left.

i. (98)16

  • 9 × 161 = 144
  • 8 × 160 = 8
  • Sum = 144 + 8 = 152
(98)16 = (152)10

ii. (1067)16

  • 1 × 163 = 4096
  • 0 × 162 = 0
  • 6 × 161 = 96
  • 7 × 160 = 7
  • Sum = 4096 + 0 + 96 + 7 = 4199
(1067)16 = (4199)10

iii. (888)16

  • 8 × 162 = 2048
  • 8 × 161 = 128
  • 8 × 160 = 8
  • Sum = 2048 + 128 + 8 = 2184
(888)16 = (2184)10

iv. (AC5)16 [A=10, C=12]

  • 10 × 162 = 2560
  • 12 × 161 = 192
  • 5 × 160 = 5
  • Sum = 2560 + 192 + 5 = 2757
(AC5)16 = (2757)10

v. (A02)16 [A=10]

  • 10 × 162 = 2560
  • 0 × 161 = 0
  • 2 × 160 = 2
  • Sum = 2560 + 0 + 2 = 2562
(A02)16 = (2562)10

vi. (FED)16 [F=15, E=14, D=13]

  • 15 × 162 = 3840
  • 14 × 161 = 224
  • 13 × 160 = 13
  • Sum = 3840 + 224 + 13 = 4077
(FED)16 = (4077)10

vii. (1E1)16 [E=14]

  • 1 × 162 = 256
  • 14 × 161 = 224
  • 1 × 160 = 1
  • Sum = 256 + 224 + 1 = 481
(1E1)16 = (481)10

viii. (8A0)16 [A=10]

  • 8 × 162 = 2048
  • 10 × 161 = 160
  • 0 × 160 = 0
  • Sum = 2048 + 160 + 0 = 2208
(8A0)16 = (2208)10

(G) Binary to Octal Conversion

Group binary digits into sets of 3 bits from right to left, then convert each group to octal.

  • i. (100)2 → 100 = 4 → (4)8
  • ii. (1010)2 → 001 | 010 = 1 | 2 → (12)8
  • iii. (11001)2 → 011 | 001 = 3 | 1 → (31)8
  • iv. (100100)2 → 100 | 100 = 4 | 4 → (44)8
  • v. (11100)2 → 011 | 100 = 3 | 4 → (34)8
  • vi. (110110)2 → 110 | 110 = 6 | 6 → (66)8
  • vii. (111110)2 → 111 | 110 = 7 | 6 → (76)8
  • viii. (11001100)2 → 011 | 001 | 100 = 3 | 1 | 4 → (314)8

(H) Octal to Binary Conversion

Convert each octal digit into its corresponding 3-bit binary code.

  • i. (64)8 → 6=110, 4=100 → (110100)2
  • ii. (104)8 → 1=001, 0=000, 4=100 → (1000100)2
  • iii. (224)8 → 2=010, 2=010, 4=100 → (10010100)2
  • iv. (333)8 → 3=011, 3=011, 3=011 → (11011011)2
  • v. (467)8 → 4=100, 6=110, 7=111 → (100110111)2
  • vi. (524)8 → 5=101, 2=010, 4=100 → (101010100)2
  • vii. (667)8 → 6=110, 6=110, 7=111 → (110110111)2
  • viii. (7004)8 → 7=111, 0=000, 0=000, 4=100 → (111000000100)2

(I) Binary to Hexadecimal Conversion

Group binary digits into sets of 4 bits from right to left, then convert to Hexadecimal.

  • i. (100)2 → 0100 = 4 → (4)16
  • ii. (1010)2 → 1010 = 10 (A) → (A)16
  • iii. (11100)2 → 0001 | 1100 = 1 | 12 (C) → (1C)16
  • iv. (110011)2 → 0011 | 0011 = 3 | 3 → (33)16
  • v. (101010)2 → 0010 | 1010 = 2 | 10 (A) → (2A)16
  • vi. (100110)2 → 0010 | 0110 = 2 | 6 → (26)16
  • vii. (11001100)2 → 1100 | 1100 = 12(C) | 12(C) → (CC)16
  • viii. (1100111)2 → 0110 | 0111 = 6 | 7 → (67)16

(J) Hexadecimal to Binary Conversion

Convert each Hexadecimal digit into its corresponding 4-bit binary code.

  • i. (81)16 → 8=1000, 1=0001 → (10000001)2
  • ii. (105)16 → 1=0001, 0=0000, 5=0101 → (100000101)2
  • iii. (981)16 → 9=1001, 8=1000, 1=0001 → (100110000001)2
  • iv. (1AC)16 → 1=0001, A=1010, C=1100 → (110101100)2
  • v. (B0F)16 → B=1011, 0=0000, F=1111 → (101100001111)2
  • vi. (BCD)16 → B=1011, C=1100, D=1101 → (101111001101)2
  • vii. (7BF)16 → 7=0111, B=1011, F=1111 → (11110111111)2
  • viii. (90E)16 → 9=1001, 0=0000, E=1110 → (100100001110)2

(K) Octal to Hexadecimal Conversion

Convert Octal to Binary first (3-bit groups), then re-group into 4-bit sets from right to left for Hexadecimal.

  • i. (74)8 → Binary: 111 100 → 0011 | 1100 → (3C)16
  • ii. (124)8 → Binary: 001 010 100 → 0101 | 0100 → (54)16
  • iii. (205)8 → Binary: 010 000 101 → 1000 | 0101 → (85)16
  • iv. (5434)8 → Binary: 101 100 011 100 → 1011 | 0001 | 1100 → (B1C)16
  • v. (1150)8 → Binary: 001 001 101 000 → 0010 | 0110 | 1000 → (268)16
  • vi. (345)8 → Binary: 011 100 101 → 1110 | 0101 → (E5)16
  • vii. (734)8 → Binary: 111 011 100 → 0001 | 1101 | 1100 → (1DC)16
  • viii. (7314)8 → Binary: 111 011 001 100 → 1110 | 1100 | 1100 → (ECC)16

(L) Hexadecimal to Octal Conversion

Convert Hexadecimal to Binary first (4-bit groups), then re-group into 3-bit sets for Octal.

  • i. (894)16 → Binary: 1000 1001 0100 → 100 | 010 | 010 | 100 → (4224)8
  • ii. (1AC)16 → Binary: 0001 1010 1100 → 000 | 110 | 101 | 100 → (654)8
  • iii. (BAD)16 → Binary: 1011 1010 1101 → 101 | 110 | 101 | 101 → (5655)8
  • iv. (10001)16 → Binary: 0001 0000 0000 0000 0001 → 000 | 010 | 000 | 000 | 000 | 001 → (200001)8
  • v. (AB6)16 → Binary: 1010 1011 0110 → 101 | 010 | 110 | 110 → (5266)8
  • vi. (B00)16 → Binary: 1011 0000 0000 → 101 | 100 | 000 | 000 → (5400)8
  • vii. (ABC)16 → Binary: 1010 1011 1100 → 101 | 010 | 111 | 100 → (5274)8

Complete Binary Arithmetic Operations: Addition, Subtraction, Multiplication, Division, and Simplification

Detailed step-by-step calculation processes for binary addition, subtraction, multiplication, division, and mixed simplification exercises.

(A) Binary Addition

Adding binary numbers according to the rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1), and 1+1+1=11 (write 1, carry 1).

i. (10100)2 + (1101)2

  • Align digits right and add column by column
  • Result with carry handling
(10100)2 + (1101)2 = (100001)2

ii. (1011)2 + (1101)2

  • Align columns and add
(1011)2 + (1101)2 = (11000)2

iii. (10111)2 + (1111)2

  • Align columns and add with carries
(10111)2 + (1111)2 = (100110)2

iv. (1110)2 + (1110)2

  • Align columns and add
(1110)2 + (1110)2 = (11100)2

v. (10100)2 + (10011)2

  • Align columns and add
(10100)2 + (10011)2 = (100111)2

vi. (111001)2 + (101100)2

  • Align columns and add with carries
(111001)2 + (101100)2 = (1100101)2

vii. (100111)2 + (11010)2

  • Align columns and add
(100111)2 + (11010)2 = (1000001)2

viii. (100100)2 + (110011)2

  • Align columns and add
(100100)2 + (110011)2 = (1010111)2

(B) Binary Subtraction

Subtracting binary numbers using borrow rules (0-0=0, 1-0=1, 1-1=0, 0-1 borrows 2 from the next column).

i. (1100)2 - (1000)2

  • Perform column subtraction
(1100)2 - (1000)2 = (100)2

ii. (1100)2 - (100)2

  • Perform column subtraction
(1100)2 - (100)2 = (1000)2

iii. (10100)2 - (110)2

  • Perform subtraction with borrowing
(10100)2 - (110)2 = (1110)2

iv. (101100)2 - (10011)2

  • Perform subtraction with borrowing
(101100)2 - (10011)2 = (11001)2

v. (1100111)2 - (1011)2

  • Perform subtraction with borrowing
(1100111)2 - (1011)2 = (1011100)2

vi. (110011)2 - (10100)2

  • Perform subtraction with borrowing
(110011)2 - (10100)2 = (11111)2

vii. (100100)2 - (11010)2

  • Perform subtraction with borrowing
(100100)2 - (11010)2 = (1010)2

viii. (1000010)2 - (110101)2

  • Perform subtraction with borrowing
(1000010)2 - (110101)2 = (1101)2

(C) Binary Multiplication

Multiplying binary numbers using partial products and binary addition.

i. (110)2 × (11)2

  • 110 × 1 = 110
  • 110 × 1 = 110 (shifted)
  • Sum = 110 + 1100 = 10010
(110)2 × (11)2 = (10010)2

ii. (101)2 × (10)2

  • Multiply by 10 (adds trailing zero)
(101)2 × (10)2 = (1010)2

iii. (111)2 × (11)2

  • 111 × 1 = 111
  • 111 × 1 = 111 (shifted)
  • Sum = 111 + 1110 = 10101
(111)2 × (11)2 = (10101)2

iv. (1100)2 × (101)2

  • 1100 × 1 = 1100
  • 1100 × 0 = 0000
  • 1100 × 1 = 1100 (shifted twice)
  • Sum = 111100
(1100)2 × (101)2 = (111100)2

v. (11010)2 × (110)2

  • Compute partial products and add
(11010)2 × (110)2 = (10011100)2

vi. (101100)2 × (100)2

  • Multiplying by 100 appends two zeros
(101100)2 × (100)2 = (10110000)2

vii. (11001)2 × (1100)2

  • Compute partial products and add
(11001)2 × (1100)2 = (100101100)2

viii. (11000)2 × (1001)2

  • Compute partial products and add
(11000)2 × (1001)2 = (11011000)2

(D) Binary Division

Dividing binary numbers to find Quotient and Remainder.

i. (111)2 ÷ (10)2

  • 7 ÷ 2 = Quotient 3 (11), Remainder 1 (1)
Quotient = (11)2, Remainder = (1)2

ii. (10110)2 ÷ (10)2

  • 22 ÷ 2 = Quotient 11 (1011), Remainder 0 (0)
Quotient = (1011)2, Remainder = (0)2

iii. (1101)2 ÷ (11)2

  • 13 ÷ 3 = Quotient 4 (100), Remainder 1 (1)
Quotient = (100)2, Remainder = (1)2

iv. (101101)2 ÷ (101)2

  • 45 ÷ 5 = Quotient 9 (1001), Remainder 0 (0)
Quotient = (1001)2, Remainder = (0)2

v. (111010)2 ÷ (110)2

  • 58 ÷ 6 = Quotient 9 (1001), Remainder 4 (100)
Quotient = (1001)2, Remainder = (100)2

vi. (100101)2 ÷ (110)2

  • 37 ÷ 6 = Quotient 6 (110), Remainder 1 (1)
Quotient = (110)2, Remainder = (1)2

vii. (1000100)2 ÷ (1100)2

  • 68 ÷ 12 = Quotient 5 (101), Remainder 8 (1000)
Quotient = (101)2, Remainder = (1000)2

viii. (1110011)2 ÷ (1010)2

  • 115 ÷ 10 = Quotient 11 (1011), Remainder 5 (101)
Quotient = (1011)2, Remainder = (101)2

(E) Binary Simplification (Mixed Operations)

Simplifying expressions containing multiple binary arithmetic operators following order of operations.

i. (11 × 10)2 + (11)2

  • Multiply: 11 × 10 = 110
  • Add: 110 + 11 = 1001
(11 × 10)2 + (11)2 = (1001)2

ii. (10 × 11)2 + (101)2

  • Multiply: 10 × 11 = 110
  • Add: 110 + 101 = 1011
(10 × 11)2 + (101)2 = (1011)2

iii. (1001 + 110)2 - (1000)2

  • Add inside parentheses: 1001 + 110 = 1111
  • Subtract: 1111 - 1000 = 0111
(1001 + 110)2 - (1000)2 = (111)2

iv. (1010 × 110)2 ÷ (10)2

  • Multiply: 1010 × 110 = 111100
  • Divide: 111100 ÷ 10 = 11110
(1010 × 110)2 ÷ (10)2 = (11110)2

v. (100110 + 110)2 ÷ (110)2

  • Add inside parentheses: 100110 + 110 = 101100
  • Divide: 101100 ÷ 110 (Quotient = 111, Remainder = 10)
Quotient = (111)2, Remainder = (10)2

vi. (10011)2 × (101)2

  • Multiply: 10011 × 101 = 1011111
(10011)2 × (101)2 = (1011111)2

vii. (1000 + 100)2 ÷ (11)2

  • Add inside parentheses: 1000 + 100 = 1100
  • Divide: 1100 ÷ 11 = 100
(1000 + 100)2 ÷ (11)2 = (100)2

viii. (11100 - 11)2 × (100)2

  • Subtract inside parentheses: 11100 - 11 = 11001
  • Multiply: 11001 × 100 = 1100100
(11100 - 11)2 × (100)2 = (1100100)2

Part 3: Class Activities & Project Work

Activity 1: Number System Summary Chart

Number System Base (Radix) Digits / Symbols Used Example
Binary 2 0, 1 (1010)2
Octal 8 0, 1, 2, 3, 4, 5, 6, 7 (157)8
Decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (95)10
Hexadecimal 16 0-9 and A-F (A=10, B=11, C=12, D=13, E=14, F=15) (3F8)16

Activity 2: Equivalence Table (Decimal 0 to 15)

Decimal (Base-10) Binary (Base-2) Octal (Base-8) Hexadecimal (Base-16)
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

Related Articles