Lesson 2 Whole Numbers
Exercise 2.1
1. Write the following numbers in expanded form:
a. 546
= 5 × 100 + 4 × 10 + 6 × 1
= 5 × 10² + 4 × 10¹ + 6 × 10⁰
b. 6542
= 6 × 1000 + 5 × 100 + 4 × 10 + 2 × 1
= 6 × 10³ + 5 × 10² + 4 × 10¹ + 2 × 10⁰
c. 1234
= 1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1
= 1 × 10³ + 2 × 10² + 3 × 10¹ + 4 × 10⁰
d. 45872
= 4 × 10000 + 5 × 1000 + 8 × 100 + 7 × 10 + 2 × 1
= 4 × 10⁴ + 5 × 10³ + 8 × 10² + 7 × 10¹ + 2 × 10⁰
e. 258963
= 2 × 100000 + 5 × 10000 + 8 × 1000 + 9 × 100 + 6 × 10 + 3 × 1
= 2 × 10⁵ + 5 × 10⁴ + 8 × 10³ + 9 × 10² + 6 × 10¹ + 3 × 10⁰
f. 97332
= 9 × 10000 + 7 × 1000 + 3 × 100 + 3 × 10 + 2 × 1
= 9 × 10⁴ + 7 × 10³ + 3 × 10² + 3 × 10¹ + 2 × 10⁰
2. Express the following numbers as the power of 2 and 5:
a. 10Power of 2:
10 = 2³ + 2¹
= 1 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰
Power of 5:
10 = 5¹ + 5⁰
= 2 × 5¹ + 0 × 5⁰
b. 25Power of 2:
25 = 2⁴ + 2³ + 2⁰
= 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
Power of 5:
25 = 5²
= 1 × 5² + 0 × 5¹ + 0 × 5⁰
c. 59Power of 2:
59 = 2⁵ + 2⁴ + 2³ + 2¹ + 2⁰
= 1 × 2⁵ + 1 × 2⁴ + 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
Power of 5:
59 = 2 × 5² + 1 × 5¹ + 4 × 5⁰
d. 402Power of 2:
402 = 2⁸ + 2⁷ + 2⁴ + 2¹
= 1 × 2⁸ + 1 × 2⁷ + 0 × 2⁶ + 0 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰
Power of 5:
402 = 3 × 5³ + 1 × 5² + 0 × 5¹ + 2 × 5⁰
e. 805Power of 2:
805 = 2⁹ + 2⁸ + 2⁵ + 2² + 2⁰
= 1 × 2⁹ + 1 × 2⁸ + 0 × 2⁷ + 0 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
Power of 5:
805 = 1 × 5⁴ + 1 × 5³ + 2 × 5² + 0 × 5¹ + 1 × 5⁰
f. 932Power of 2:
932 = 2⁹ + 2⁸ + 2⁷ + 2⁵ + 2¹
= 1 × 2⁹ + 1 × 2⁸ + 1 × 2⁷ + 0 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 1 × 2¹ + 0 × 2⁰
Power of 5:
932 = 1 × 5⁴ + 2 × 5³ + 2 × 5² + 0 × 5¹ + 1 × 5⁰
3. State the number system of the following numbers:
a. 10011₂ - Binary
b. 350 - Decimal
c. 1001₂ - Binary
d. 42₅ - Quinary
e. 555 - Decimal
f. 77532 - Decimal
g. 10010011₂ - Binary
h. 257903 - Decimal
i. 4023₅ - Quinary
j. 5321 - Decimal
k. 1234₅ - Quinary
l. 1010111₂ - Binary
4. Convert the following numbers from the decimal number system to the binary number system:
a. 4
Solution:
4 ÷ 2 = 2, remainder 0
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
4 = 100₂
b. 9
Solution:
9 ÷ 2 = 4, remainder 1
4 ÷ 2 = 2, remainder 0
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
9 = 1001₂
c. 12
Solution:
12 ÷ 2 = 6, remainder 0
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
12 = 1100₂
d. 25
Solution:
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
Reading the remainders from bottom to top:
25 = 11001₂
e. 35
Solution:
35 ÷ 2 = 17, remainder 1
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
Reading the remainders from bottom to top:
35 = 100011₂
f. 79
Solution:
79 ÷ 2 = 39, remainder 1
39 ÷ 2 = 19, remainder 1
19 ÷ 2 = 9, remainder 1
9 ÷ 2 = 4, remainder 1
4 ÷ 2 = 2, remainder 0
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
79 = 1001111₂
g. 94
Solution:
94 ÷ 2 = 47, remainder 0
47 ÷ 2 = 23, remainder 1
23 ÷ 2 = 11, remainder 1
11 ÷ 2 = 5, remainder 1
5 ÷ 2 = 2, remainder 1
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
94 = 1011110₂
h. 100
Solution:
100 ÷ 2 = 50, remainder 0
50 ÷ 2 = 25, remainder 0
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
Reading the remainders from bottom to top:
100 = 1100100₂
i. 104
Solution:
104 ÷ 2 = 52, remainder 0
52 ÷ 2 = 26, remainder 0
26 ÷ 2 = 13, remainder 0
13 ÷ 2 = 6, remainder 1
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
104 = 1101000₂
j. 135
Solution:
135 ÷ 2 = 67, remainder 1
67 ÷ 2 = 33, remainder 1
33 ÷ 2 = 16, remainder 1
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
Reading the remainders from bottom to top:
135 = 10000111₂
k. 190
Solution:
190 ÷ 2 = 95, remainder 0
95 ÷ 2 = 47, remainder 1
47 ÷ 2 = 23, remainder 1
23 ÷ 2 = 11, remainder 1
11 ÷ 2 = 5, remainder 1
5 ÷ 2 = 2, remainder 1
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
190 = 10111110₂
l. 22
Solution:
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
Reading the remainders from bottom to top:
22 = 10110₂
m. 250
Solution:
250 ÷ 2 = 125, remainder 0
125 ÷ 2 = 62, remainder 1
62 ÷ 2 = 31, remainder 0
31 ÷ 2 = 15, remainder 1
15 ÷ 2 = 7, remainder 1
7 ÷ 2 = 3, remainder 1
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top:
250 = 11111010₂
n. 275
Solution:
275 ÷ 2 = 137, remainder 1
137 ÷ 2 = 68, remainder 1
68 ÷ 2 = 34, remainder 0
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
Reading the remainders from bottom to top:
275 = 100010011₂
o. 366
Solution:
366 ÷ 2 = 183, remainder 0
183 ÷ 2 = 91, remainder 1
91 ÷ 2 = 45, remainder 1
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
Reading the remainders from bottom to top:
366 = 101101110₂
p. 512
Solution:
512 ÷ 2 = 256, remainder 0
256 ÷ 2 = 128, remainder 0
128 ÷ 2 = 64, remainder 0
5. Convert the following numbers from the binary number system to the decimal number system.
a. 11₂
= 1 × 2¹ + 1 × 2⁰
= 2 + 1
= 3
Decimal result: 3
b. 101₂
= 1 × 2² + 0 × 2¹ + 1 × 2⁰
= 4 + 0 + 1
= 5
Decimal result: 5
c. 111₂
= 1 × 2² + 1 × 2¹ + 1 × 2⁰
= 4 + 2 + 1
= 7
Decimal result: 7
d. 1100₂
= 1 × 2³ + 1 × 2² + 0 × 2¹ + 0 × 2⁰
= 8 + 4 + 0 + 0
= 12
Decimal result: 12
e. 10101₂
= 1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
= 16 + 0 + 4 + 0 + 1
= 21
Decimal result: 21
f. 11001₂
= 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
= 16 + 8 + 0 + 0 + 1
= 25
Decimal result: 25
g. 10010₂
= 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰
= 16 + 0 + 0 + 2 + 0
= 18
Decimal result: 18
h. 11110₂
= 1 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰
= 16 + 8 + 4 + 2 + 0
= 30
Decimal result: 30
i. 100001₂
= 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
= 32 + 0 + 0 + 0 + 0 + 1
= 33
Decimal result: 33
j. 111111₂
= 1 × 2⁵ + 1 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰
= 32 + 16 + 8 + 4 + 2 + 1
= 63
Decimal result: 63
k. 1100011₂
= 1 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
= 64 + 32 + 0 + 0 + 0 + 2 + 1
= 99
Decimal result: 99
l. 1110011₂
= 1 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
= 64 + 32 + 16 + 0 + 0 + 2 + 1
= 115
Decimal result: 115
m. 1100110011₂
= 1 × 2⁹ + 1 × 2⁸ + 0 × 2⁷ + 0 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
= 512 + 256 + 0 + 0 + 32 + 16 + 0 + 0 + 2 + 1
= 819
Decimal result: 819
n. 1010101110₂
= 1 × 2⁹ + 0 × 2⁸ + 1 × 2⁷ + 0 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰
= 512 + 0 + 128 + 0 + 32 + 0 + 8 + 4 + 2 + 0
= 686
Decimal result: 686
o. 100001000₂
= 1 × 2⁸ + 0 × 2⁷ + 0 × 2⁶ + 0 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 0 × 2⁰
= 256 + 0 + 0 + 0 + 0 + 8 + 0 + 0 + 0
Decimal result: 264
p. 101110111₂
= 1 × 2⁸ + 0 × 2⁷ + 1 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰
= 256 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 1
Decimal result: 375
q. 11011011001₂
= 1 × 2¹⁰ + 1 × 2⁹ + 0 × 2⁸ + 1 × 2⁷ + 1 × 2⁶ + 0 × 2⁵ + 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
= 1024 + 512 + 0 + 128 + 64 + 0 + 16 + 8 + 0 + 0 + 1
Decimal result: 1753
r. 1111111110₂
= 1 × 2⁹ + 1 × 2⁸ + 1 × 2⁷ + 1 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰
= 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0
Decimal result: 1022
6. Convert the following numbers to the quinary number system.
a. 9 ÷ 5 = 1, remainder 4
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
9 = 14₅
b. 13 ÷ 5 = 2, remainder 3
2 ÷ 5 = 0, remainder 2
Reading the remainders from bottom to top:
13 = 23₅
c. 21 ÷ 5 = 4, remainder 1
4 ÷ 5 = 0, remainder 4
Reading the remainders from bottom to top:
21 = 41₅
d. 26 ÷ 5 = 5, remainder 1
5 ÷ 5 = 1, remainder 0
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
26 = 101₅
e. 45 ÷ 5 = 9, remainder 0
9 ÷ 5 = 1, remainder 4
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
45 = 140₅
f. 55 ÷ 5 = 11, remainder 0
11 ÷ 5 = 2, remainder 1
2 ÷ 5 = 0, remainder 2
Reading the remainders from bottom to top:
55 = 210₅
g. 86 ÷ 5 = 17, remainder 1
17 ÷ 5 = 3, remainder 2
3 ÷ 5 = 0, remainder 3
Reading the remainders from bottom to top:
86 = 321₅
h. 112 ÷ 5 = 22, remainder 2
22 ÷ 5 = 4, remainder 2
4 ÷ 5 = 0, remainder 4
Reading the remainders from bottom to top:
112 = 422₅
i. 194 ÷ 5 = 38, remainder 4
38 ÷ 5 = 7, remainder 3
7 ÷ 5 = 1, remainder 2
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
194 = 1324₅
j. 404 ÷ 5 = 80, remainder 4
80 ÷ 5 = 16, remainder 0
16 ÷ 5 = 3, remainder 1
3 ÷ 5 = 0, remainder 3
Reading the remainders from bottom to top:
404 = 3104₅
k. 497 ÷ 5 = 99, remainder 2
99 ÷ 5 = 19, remainder 4
19 ÷ 5 = 3, remainder 4
3 ÷ 5 = 0, remainder 3
Reading the remainders from bottom to top:
497 = 3442₅
l. 650 ÷ 5 = 130, remainder 0
130 ÷ 5 = 26, remainder 0
26 ÷ 5 = 5, remainder 1
5 ÷ 5 = 1, remainder 0
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
650 = 10100₅
m. 1128 ÷ 5 = 225, remainder 3
225 ÷ 5 = 45, remainder 0
45 ÷ 5 = 9, remainder 0
9 ÷ 5 = 1, remainder 4
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
1128 = 14003₅
n. 1234 ÷ 5 = 246, remainder 4
246 ÷ 5 = 49, remainder 1
49 ÷ 5 = 9, remainder 4
9 ÷ 5 = 1, remainder 4
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
1234 = 14414₅
o. 2125 ÷ 5 = 425, remainder 0
425 ÷ 5 = 85, remainder 0
85 ÷ 5 = 17, remainder 0
17 ÷ 5 = 3, remainder 2
3 ÷ 5 = 0, remainder 3
Reading the remainders from bottom to top:
2125 = 32000₅
p. 2536 ÷ 5 = 507, remainder 1
507 ÷ 5 = 101, remainder 2
101 ÷ 5 = 20, remainder 1
20 ÷ 5 = 4, remainder 0
4 ÷ 5 = 0, remainder 4
Reading the remainders from bottom to top:
2536 = 40121₅
q. 3000 ÷ 5 = 600, remainder 0
600 ÷ 5 = 120, remainder 0
120 ÷ 5 = 24, remainder 0
24 ÷ 5 = 4, remainder 4
4 ÷ 5 = 0, remainder 4
Reading the remainders from bottom to top:
3000 = 44000₅
r. 3650 ÷ 5 = 730, remainder 0
730 ÷ 5 = 146, remainder 0
146 ÷ 5 = 29, remainder 1
29 ÷ 5 = 5, remainder 4
5 ÷ 5 = 1, remainder 0
1 ÷ 5 = 0, remainder 1
Reading the remainders from bottom to top:
3650 = 10410₅
7. Convert the following numbers from the quinary number system to the decimal number system.
a. 21₅ = 2 × 5¹ + 1 × 5⁰ = 10 + 1 = 11
b. 24₅ = 2 × 5¹ + 4 × 5⁰ = 10 + 4 = 14
c. 34₅ = 3 × 5¹ + 4 × 5⁰ = 15 + 4 = 19
d. 101₅ = 1 × 5² + 0 × 5¹ + 1 × 5⁰ = 25 + 0 + 1 = 26
e. 123₅ = 1 × 5² + 2 × 5¹ + 3 × 5⁰ = 25 + 10 + 3 = 38
f. 300₅ = 3 × 5² + 0 × 5¹ + 0 × 5⁰ = 75 + 0 + 0 = 75
g. 343₅ = 3 × 5² + 4 × 5¹ + 3 × 5⁰ = 75 + 20 + 3 = 98
h. 441₅ = 4 × 5² + 4 × 5¹ + 1 × 5⁰ = 100 + 20 + 1 = 121
i. 2023₅ = 2 × 5³ + 0 × 5² + 2 × 5¹ + 3 × 5⁰ = 250 + 0 + 10 + 3 = 263
j. 1234₅ = 1 × 5³ + 2 × 5² + 3 × 5¹ + 4 × 5⁰ = 125 + 50 + 15 + 4 = 194
k. 2113₅ = 2 × 5³ + 1 × 5² + 1 × 5¹ + 3 × 5⁰ = 250 + 25 + 5 + 3 = 283
l. 3313₅ = 3 × 5³ + 3 × 5² + 1 × 5¹ + 3 × 5⁰ = 375 + 75 + 5 + 3 = 458
m. 2014₅ = 2 × 5³ + 0 × 5² + 1 × 5¹ + 4 × 5⁰ = 250 + 0 + 5 + 4 = 259
n. 4201₅ = 4 × 5³ + 2 × 5² + 0 × 5¹ + 1 × 5⁰ = 500 + 50 + 0 + 1 = 551
o. 4321₅ = 4 × 5³ + 3 × 5² + 2 × 5¹ + 1 × 5⁰ = 500 + 75 + 10 + 1 = 586
p. 12304₅ = 1 × 5⁴ + 2 × 5³ + 3 × 5² + 0 × 5¹ + 4 × 5⁰ = 625 + 250 + 75 + 0 + 4 = 954
q. 10123₅ = 1 × 5⁴ + 0 × 5³ + 1 × 5² + 2 × 5¹ + 3 × 5⁰ = 625 + 0 + 25 + 10 + 3 = 663
r. 21432₅ = 2 × 5⁴ + 1 × 5³ + 4 × 5² + 3 × 5¹ + 2 × 5⁰ = 1250 + 125 + 100 + 15 + 2 = 1492
8. Convert the following to binary if it is in quinary and quinary if it is in binary number system.
a. 40₅ = 4 × 5¹ + 0 × 5⁰ = 20 + 0 = 20
20 ÷ 2 = 10, remainder 0
10 ÷ 2 = 5, remainder 0
5 ÷ 2 = 2, remainder 1
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 20 = 10100₂
b. 101012₅ = 1 × 5⁵ + 0 × 5⁴ + 1 × 5³ + 0 × 5² + 1 × 5¹ + 2 × 5⁰ = 3125 + 0 + 125 + 0 + 5 + 2 = 3257
3257 ÷ 2 = 1628, remainder 1
1628 ÷ 2 = 814, remainder 0
814 ÷ 2 = 407, remainder 0
407 ÷ 2 = 203, remainder 1
203 ÷ 2 = 101, remainder 1
101 ÷ 2 = 50, remainder 1
50 ÷ 2 = 25, remainder 0
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
Reading the remainders from bottom to top: 3257 = 110010101001₂
c. 31₅ = 3 × 5¹ + 1 × 5⁰ = 15 + 1 = 16
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
Reading the remainders from bottom to top: 16 = 10000₂
d. 1012₅ = 1 × 5³ + 0 × 5² + 1 × 5¹ + 2 × 5⁰ = 125 + 0 + 5 + 2 = 132
132 ÷ 2 = 66, remainder 0
66 ÷ 2 = 33, remainder 0
33 ÷ 2 = 16, remainder 1
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
Reading the remainders from bottom to top: 132 = 10000100₂
e. 144₅ = 1 × 5² + 4 × 5¹ + 4 × 5⁰ = 25 + 20 + 4 = 49
49 ÷ 2 = 24, remainder 1
24 ÷ 2 = 12, remainder 0
12 ÷ 2 = 6, remainder 0
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 49 = 110001₂
f. 23₅ = 2 × 5¹ + 3 × 5⁰ = 10 + 3 = 13
13 ÷ 2 = 6, remainder 1
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 13 = 1101₂
g. 143₅ = 1 × 5² + 4 × 5¹ + 3 × 5⁰ = 25 + 20 + 3 = 48
48 ÷ 2 = 24, remainder 0
24 ÷ 2 = 12, remainder 0
12 ÷ 2 = 6, remainder 0
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 48 = 110000₂
a. Which of the numbers 1101₂ and 24₅ is larger? Also, find their difference.
Solution:
Comparing 1101₂ and 24₅:
First, convert both numbers to decimal.
1101₂ = 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
= 8 + 4 + 0 + 1
= 13 (decimal)
24₅ = 2 × 5¹ + 4 × 5⁰
= 10 + 4
= 14 (decimal)
Now compare: 14 > 13, so 24₅ is larger.
Difference: 14 - 13 = 1 (decimal).
Final answer:
The number 24₅ is larger than 1101₂, and their difference is 1.
b. Which of the numbers 110011₂ and 144₅ is larger? Also, find their difference.
Solution:
Comparing 110011₂ and 144₅:
First, convert both numbers to decimal.
110011₂ = 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
= 32 + 16 + 0 + 0 + 2 + 1
= 51 (decimal)
144₅ = 1 × 5² + 4 × 5¹ + 4 × 5⁰
= 25 + 20 + 4
= 49 (decimal)
Now compare: 51 > 49, so 110011₂ is larger.
Difference: 51 - 49 = 2 (decimal).
Final answer:
The number 110011₂ is larger than 144₅, and their difference is 2.
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