Number Systems — Complete Solutions

🔑 Quick Tricks

Decimal→Binary: 2 se divide karo

Decimal→Octal: 8 se divide karo

Decimal→Hex: 16 se divide karo

Binary→Decimal: 2 ki power multiply

Hex digits: 0-9, A=10, B=11, C=12, D=13, E=14, F=15

Section E — Theory

Answer the Following Questions 📖

Binary

Base = 2

Digits: 0, 1

Octal

Base = 8

Digits: 0–7

Decimal

Base = 10

Digits: 0–9

Hexadecimal

Base = 16

Digits: 0–9, A–F

What is a number system?

Answer A number system is a way of representing numbers using a set of symbols (called digits) and a base. It defines how we write, count, and do math with numbers.

💡 Simple bolein toh: Number system ek language hai numbers ke liye. Jaise humans ke liye 0–9 use karte hain (decimal), waise computers ke liye sirf 0 aur 1 (binary).

What is the base of a number system? Why is it important?

Answer The base (or radix) of a number system is the total count of unique digits used in that system.

Examples: Binary → base 2 (uses 0,1) | Octal → base 8 | Decimal → base 10 | Hex → base 16

Why Important? The base tells us the value of each digit's position. In decimal, positions are powers of 10 (ones, tens, hundreds). In binary, positions are powers of 2. Knowing the base is essential to convert between systems correctly.

💡 Base = kitne digits available hain us system mein. Isse hi pata chalta hai har digit ki positional value kya hai!

What does "hexadecimal" mean in the context of number systems?

Answer Hexadecimal means a number system with base 16. It uses 16 unique symbols: digits 0–9 and letters A, B, C, D, E, F (where A=10, B=11, C=12, D=13, E=14, F=15).

Why used in computers? Binary numbers are very long. Hexadecimal compresses them — 4 binary digits = 1 hex digit. So programmers use hex to represent binary data more compactly. Example: 1111 1111 in binary = FF in hex.

💡 "Hexa" = 6, "Decimal" = 10, toh 6+10 = 16 digits! Hex mein A se F tak letters bhi digits ka kaam karte hain.

Why do we need to convert a decimal number into binary?

Answer We convert decimal to binary because computers only understand binary (0 and 1). Every piece of data — text, images, audio — is stored and processed as binary inside a computer. The 0 represents OFF state and 1 represents ON state of electronic switches (transistors).

💡 Computer ek electronic machine hai. Uske andar sirf 2 states hain: current ON (1) ya current OFF (0). Isliye sab kuch binary mein convert karna padta hai!

Section F — Question 1

Decimal → Octal, Hexadecimal, Binary 🔄

Method Yaad Karo Binary: Number ko 2 se baar baar divide karo, remainders ko ulta likhkar answer milta hai.
Octal: 8 se divide karo same tarike se.
Hex: 16 se divide karo. Agar remainder 10 hai toh A, 11→B, 12→C, 13→D, 14→E, 15→F.

Decimal	Binary (Base 2)	Octal (Base 8)	Hexadecimal (Base 16)
(12)₁₀	1100	14	C
(89)₁₀	1011001	131	59
(361)₁₀	101101001	551	169
(671)₁₀	1010011111	1237	29F
(906)₁₀	1110001010	1612	38A

Detailed steps for (12)₁₀ → Binary, Octal, Hex

Binary (÷2)

12 ÷ 2 = 6 R 0
6 ÷ 2 = 3 R 0
3 ÷ 2 = 1 R 1
1 ÷ 2 = 0 R 1

↑ Read up: 1100

Octal (÷8)

12 ÷ 8 = 1 R 4
1 ÷ 8 = 0 R 1

↑ Read up: 14

Hex (÷16)

12 ÷ 16 = 0 R 12
12 = C in hex

↑ Read up: C

✅ (12)₁₀ = (1100)₂ = (14)₈ = (C)₁₆

Detailed steps for (89)₁₀ → Binary, Octal, Hex

Binary (÷2)

89 ÷ 2 = 44 R 1
44 ÷ 2 = 22 R 0
22 ÷ 2 = 11 R 0
11 ÷ 2 = 5 R 1
5 ÷ 2 = 2 R 0
2 ÷ 2 = 1 R 0
1 ÷ 2 = 0 R 1

↑ 1011001

Octal (÷8)

89 ÷ 8 = 11 R 1
11 ÷ 8 = 1 R 3
1 ÷ 8 = 0 R 1

↑ 131

Hex (÷16)

89 ÷ 16 = 5 R 9
5 ÷ 16 = 0 R 5

↑ 59

✅ (89)₁₀ = (1011001)₂ = (131)₈ = (59)₁₆

Detailed steps for (361)₁₀ → Binary, Octal, Hex

Binary (÷2)

361÷2=180 R 1
180÷2=90 R 0
90÷2=45 R 0
45÷2=22 R 1
22÷2=11 R 0
11÷2=5 R 1
5÷2=2 R 1
2÷2=1 R 0
1÷2=0 R 1

↑ 101101001

Octal (÷8)

361÷8=45 R 1
45÷8=5 R 5
5÷8=0 R 5

↑ 551

Hex (÷16)

361÷16=22 R 9
22÷16=1 R 6
1÷16=0 R 1

↑ 169

✅ (361)₁₀ = (101101001)₂ = (551)₈ = (169)₁₆

Detailed steps for (671)₁₀ → Binary, Octal, Hex

Binary (÷2)

671÷2=335 R 1
335÷2=167 R 1
167÷2=83 R 1
83÷2=41 R 1
41÷2=20 R 1
20÷2=10 R 0
10÷2=5 R 0
5÷2=2 R 1
2÷2=1 R 0
1÷2=0 R 1

↑ 1010011111

Octal (÷8)

671÷8=83 R 7
83÷8=10 R 3
10÷8=1 R 2
1÷8=0 R 1

↑ 1237

Hex (÷16)

671÷16=41 R 15=F
41÷16=2 R 9
2÷16=0 R 2

↑ 29F

✅ (671)₁₀ = (1010011111)₂ = (1237)₈ = (29F)₁₆

Detailed steps for (906)₁₀ → Binary, Octal, Hex

Binary (÷2)

906÷2=453 R 0
453÷2=226 R 1
226÷2=113 R 0
113÷2=56 R 1
56÷2=28 R 0
28÷2=14 R 0
14÷2=7 R 0
7÷2=3 R 1
3÷2=1 R 1
1÷2=0 R 1

↑ 1110001010

Octal (÷8)

906÷8=113 R 2
113÷8=14 R 1
14÷8=1 R 6
1÷8=0 R 1

↑ 1612

Hex (÷16)

906÷16=56 R 10=A
56÷16=3 R 8
3÷16=0 R 3

↑ 38A

✅ (906)₁₀ = (1110001010)₂ = (1612)₈ = (38A)₁₆

Section F — Question 2

Binary → Decimal, Octal, Hexadecimal 🔄

Method — Binary to Decimal Har bit ko uski position ki power of 2 se multiply karo (right se start: 2⁰, 2¹, 2², ...) aur sab add karo.
Then: Decimal milne ke baad, usse 8 ya 16 se divide karke Octal/Hex nikalo.

Binary	Decimal (Base 10)	Octal (Base 8)	Hexadecimal (Base 16)
(11)₂	3	3	3
(110)₂	6	6	6
(100011)₂	35	43	23
(111010)₂	58	72	3A
(10001001)₂	137	211	89

Click to see detailed step-by-step for all F2 parts

110

100011

111010

10001001

(11)₂ → Decimal: 1×2¹ + 1×2⁰ = 2 + 1 = 3
3 → Octal: 3÷8 = 0 R 3 → (3)₈
3 → Hex: 3÷16 = 0 R 3 → (3)₁₆

✅ (11)₂ = (3)₁₀ = (3)₈ = (3)₁₆

(110)₂ → Decimal: 1×2² + 1×2¹ + 0×2⁰ = 4+2+0 = 6
6 → Octal: 6÷8 = 0 R 6 → (6)₈
6 → Hex: 6÷16 = 0 R 6 → (6)₁₆

✅ (110)₂ = (6)₁₀ = (6)₈ = (6)₁₆

(100011)₂ → Decimal:
1×2⁵ + 0×2⁴ + 0×2³ + 0×2² + 1×2¹ + 1×2⁰
= 32 + 0 + 0 + 0 + 2 + 1 = 35
35 → Octal: 35÷8=4 R 3, 4÷8=0 R 4 → read up: (43)₈
35 → Hex: 35÷16=2 R 3, 2÷16=0 R 2 → read up: (23)₁₆

✅ (100011)₂ = (35)₁₀ = (43)₈ = (23)₁₆

(111010)₂ → Decimal:
1×2⁵+1×2⁴+1×2³+0×2²+1×2¹+0×2⁰
= 32+16+8+0+2+0 = 58
58 → Octal: 58÷8=7 R 2, 7÷8=0 R 7 → (72)₈
58 → Hex: 58÷16=3 R 10=A, 3÷16=0 R 3 → (3A)₁₆

✅ (111010)₂ = (58)₁₀ = (72)₈ = (3A)₁₆

(10001001)₂ → Decimal:
1×2⁷+0×2⁶+0×2⁵+0×2⁴+1×2³+0×2²+0×2¹+1×2⁰
= 128+0+0+0+8+0+0+1 = 137
137 → Octal: 137÷8=17 R 1, 17÷8=2 R 1, 2÷8=0 R 2 → (211)₈
137 → Hex: 137÷16=8 R 9, 8÷16=0 R 8 → (89)₁₆

✅ (10001001)₂ = (137)₁₀ = (211)₈ = (89)₁₆

Section F — Question 3

Hex/Octal → Decimal & Binary 🔄

Method Octal→Decimal: Har digit × 8 ki power (right se 8⁰, 8¹, 8²...) add karo.
Hex→Decimal: Har digit × 16 ki power. A=10, B=11, C=12, D=13, E=14, F=15.
→Binary: Decimal milne ke baad 2 se divide karo.

Number	Decimal	Binary
(1B)₁₆	27	11011
(561)₈	369	101110001
(717)₈	463	111001111
(B28)₁₆	2856	101100101000
(D4E)₁₆	3406	110101001110

Click to see detailed steps for all F3 parts

561₈

717₈

B28

D4E

(1B)₁₆ → Decimal: B=11
1×16¹ + 11×16⁰ = 16 + 11 = 27
27 → Binary: 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
Read up: 11011

✅ (1B)₁₆ = (27)₁₀ = (11011)₂

(561)₈ → Decimal:
5×8²+6×8¹+1×8⁰ = 5×64+6×8+1 = 320+48+1 = 369
369 → Binary: 369÷2=184 R1, 184÷2=92 R0, 92÷2=46 R0, 46÷2=23 R0, 23÷2=11 R1, 11÷2=5 R1, 5÷2=2 R1, 2÷2=1 R0, 1÷2=0 R1
Read up: 101110001

✅ (561)₈ = (369)₁₀ = (101110001)₂

(717)₈ → Decimal:
7×8²+1×8¹+7×8⁰ = 7×64+8+7 = 448+8+7 = 463
463 → Binary: 463÷2=231 R1, 231÷2=115 R1, 115÷2=57 R1, 57÷2=28 R1, 28÷2=14 R0, 14÷2=7 R0, 7÷2=3 R1, 3÷2=1 R1, 1÷2=0 R1
Read up: 111001111

✅ (717)₈ = (463)₁₀ = (111001111)₂

(B28)₁₆ → Decimal: B=11
11×16²+2×16¹+8×16⁰ = 11×256+32+8 = 2816+32+8 = 2856
2856 → Binary: Divide by 2 repeatedly:
2856→1428 R0, 1428→714 R0, 714→357 R0, 357→178 R1, 178→89 R0, 89→44 R1, 44→22 R0, 22→11 R0, 11→5 R1, 5→2 R1, 2→1 R0, 1→0 R1
Read up: 101100101000

✅ (B28)₁₆ = (2856)₁₀ = (101100101000)₂

(D4E)₁₆ → Decimal: D=13, E=14
13×16²+4×16¹+14×16⁰ = 13×256+64+14 = 3328+64+14 = 3406
3406 → Binary: 3406→1703 R0, 1703→851 R1, 851→425 R1, 425→212 R1, 212→106 R0, 106→53 R0, 53→26 R1, 26→13 R0, 13→6 R1, 6→3 R0, 3→1 R1, 1→0 R1
Read up: 110101001110

✅ (D4E)₁₆ = (3406)₁₀ = (110101001110)₂

Section F — Question 4

Add Binary Numbers ➕

Binary Addition Rules — Yaad karo! 0+0 = 0 | 0+1 = 1 | 1+0 = 1 | 1+1 = 10 (write 0, carry 1) | 1+1+1 = 11 (write 1, carry 1)

(101)₂ + (110)₂

Column Addition

1 0 1
+ 1 1 0

1 0 1 1

Step by step

Col 1 (right): 1+0=1 ✓
Col 2: 0+1=1 ✓
Col 3: 1+1=10 → write 0, carry 1
Col 4: carry 1 → write 1

✅ (101)₂ + (110)₂ = (1011)₂

💡 Decimal mein check karo: 5+6=11 aur (1011)₂ = 11. Sahi hai! ✓

(10101)₂ + (1010)₂

Column Addition

1 0 1 0 1
+ 0 1 0 1 0

1 1 1 1 1

Step by step

Col1: 1+0=1
Col2: 0+1=1
Col3: 1+0=1
Col4: 0+1=1
Col5: 1+0=1

✅ (10101)₂ + (1010)₂ = (11111)₂

💡 Decimal check: 21+10=31, (11111)₂=31 ✓

(1110101)₂ + (110110)₂

Column Addition

carries: 1 1 1 1 0 0 0
1 1 1 0 1 0 1
+ 0 1 1 0 1 1 0

1 0 1 0 1 0 1 1

Step by step

Col1: 1+0=1
Col2: 0+1=1, c=0
Col3: 1+1=10 → write 0, carry 1
Col4: 0+0+c1=1
Col5: 1+1=10 → write 0, carry 1
Col6: 1+1+c1=11 → write 1, carry 1
Col7: 1+0+c1=10 → write 0, carry 1
Col8: carry 1 → 1

✅ (1110101)₂ + (110110)₂ = (10101011)₂

💡 Decimal check: 117+54=171, (10101011)₂=171 ✓

Section F — Question 5

Subtract Binary Numbers ➖

Binary Subtraction Rules — Yaad karo! 0−0 = 0 | 1−0 = 1 | 1−1 = 0 | 0−1 = borrow karo → 10−1=1 (aur agle column se 1 minus ho jaata hai)

(110)₂ − (11)₂ [Second minus First]

Subtraction

1 1 0
− 0 1 1

0 1 1

Step by step

Col1: 0−1 → borrow! 10−1=1, next col mein 1 minus
Col2: 1−1−1(borrow)=−1 → borrow again! 11−1−1=1
Col3: 1−0−1(borrow)=0

✅ (110)₂ − (11)₂ = (11)₂

💡 Decimal check: 6−3=3 aur (11)₂=3 ✓

(10101)₂ − (1010)₂ [Second minus First]

Subtraction

1 0 1 0 1
− 0 1 0 1 0

0 1 0 1 1

Step by step

Col1: 1−0=1
Col2: 0−1 → borrow! 10−1=1, next−1
Col3: 1−0−1(borrow)=0
Col4: 0−1 → borrow! 10−1=1, next−1
Col5: 1−0−1(borrow)=0

✅ (10101)₂ − (1010)₂ = (1011)₂

💡 Decimal check: 21−10=11 aur (1011)₂=11 ✓

(1110101)₂ − (110110)₂ [Second minus First]

Subtraction

1 1 1 0 1 0 1
− 0 1 1 0 1 1 0

0 1 1 1 1 1 1

Step by step

Col1: 1−0=1
Col2: 0−1 → borrow! 10−1=1, next−1
Col3: 1−1−1=−1 → borrow! 11−1−1=1, next−1
Col4: 0−0−1=−1 → borrow! 10−0−1=1, next−1
Col5: 1−1−1=−1 → borrow! 11−1−1=1, next−1
Col6: 1−1−1=−1 → borrow! 11−1−1=1, next−1
Col7: 1−0−1=0

✅ (1110101)₂ − (110110)₂ = (111111)₂

💡 Decimal check: 117−54=63 aur (111111)₂=63 ✓