Ch 1 Numbers – Exercises (2024)

PART B

I. Convert the following to their binary equivalents:

(784)10 = (1100001000)2
(999)10 = (1111100111)2
(1401)10 = (10101110001)2
(1423)10 = (10110001111)2
(478.75)10 = (111011110.11)2
(165.35)10 = (10100101.01)2
(277.27)10 = (100010101.01)2
(322.2)10 = (101000010.10)2

II. Perform the following:

(1010110)2 = (86)10
(1000011)2 = (67)10
(1100111)2 = (103)10
(10101011)2 = (171)10
(1010.001)2 = (10.125)10
(10100.11)2 = (20.75)10
(1010.111)2 = (10.875)10
(1010.0111)2 = (10.4375)10

III. Add the following binary numbers:

(1101111)2 + (1011011)2 = (10010010)2
(1111011)2 – (1011101)2 = (1000110)2
(10111)2 + (110111)2 = (1001000)2
(1101101)2 + (1110010)2 = (10111111)2
(10111101)2 + (1101101)2 = (100100000)2

IV. Subtract the following:

(110101)2 – (011010)2 = (011011)2
(110110)2 – (01010)2 = (100100)2
(101011)2 – (11101)2 = (10010)2
(11111)2 – (01101)2 = (10010)2
(10000)2 – (01111)2 = (00001)2

V. Perform the following arithmetic:

(1546)8 + (9641)8 = (2027)8
(5142)8 – (2546)8 = (2576)8
(1F5C)16 + (AC2B)16 = (2C87)16
(D53F)16 – (A68D)16 = (1E52)16
(EA9C)16 – (ABCD)16 = (3F0F)16

VI. Perform the following multiplication:

(10110)2 × (110)2 = (1111110)2
(10101)2 × (100)2 = (1010100)2
(11000)2 × (101)2 = (1111000)2
(10.001)2 × (0.11)2 = (1.100011)2
(101.10)2 × (1.10)2 = (110.1110)2
(10.101)2 × (10.1)2 = (110.1101)2
(100.01)2 × (11.1)2 = (1111.111)2
(101.001)2 × (0.11)2 = (11.100011)2

VII. Convert to Octal equivalents:

(4507)10 = (10613)8
(7757)10 = (16615)8
(7606)10 = (16516)8
(10110111)2 = (267)8
(110101101)2 = (325)8
(100001111)2 = (207)8
(11010.0101)2 = (32.05)8
(1000.011)2 = (10.03)8
(451.125)10 = (673.2)8
(245.53)10 = (365.4)8
(430.26)10 = (654.4)8
(242.24)10 = (362.3)8

VIII. Convert the following to Decimal numbers:

(5100)8 = (2624)10
(7070)8 = (3592)10
(4720)8 = (2512)10
(A452)16 = (42082)10
(ABC)16 = (43996)10
(CD7)16 = (32855)10
(27.64)8 = (23.5)10
(57.55)8 = (47.4375)10
(301.26)8 = (193.1875)10
(707.71)8 = (459.5625)10

IX. Perform the following:

(3402)8 = (1434)2
(1507)8 = (651)2
(2003)8 = (1001)2
(ABC)16 = (101010111100)2
(9AD)16 = (100110101101)2
(DE)16 = (11011110)2
(345.33)8 = (11100101.010011)2
(501.75)8 = (10011101.110011)2
(265.55)8 = (10100101.101101)2
(334.33)8 = (11011100.010011)2

X. Convert the following to their Hexa-decimal equivalents:

(11001110111)2 = (19F7)16
(100101101110)2 = (12F6)16
(11010111100)2 = (1AF4)16
(9894)10 = (269E)16
(89392)10 = (15D30)16
(4966)10 = (1376)16
(11001110.00100111)2 = (1CE.25)16
(10010110.00111110)2 = (12E.7E)16
(1554.115)8 = (3C6.29)16
(7013.2011)8 = (1C2D.41)16

XI. Convert the following to their Binary equivalents followed by Octal equivalents:

(ABC)16 = (101010111100)2 = (1254)8
(2CDE)16 = (101100111011110)2 = (1356)8
(B45)16 = (101101000101)2 = (1645)8
(4DC3)16 = (10011101100011)2 = (2353)8
(786A)16 = (1111000110101010)2 = (1642)8
(2345)16 = (10001101000101)2 = (1055)8

PART C

I. Answer the following questions:

Give a brief concept of counting system of primitive people.

Primitive people used various methods to count, such as using fingers, toes, and other body parts. They also used objects like stones, sticks, and shells to represent numbers.

What do you understand by Decimal Odometer?

A decimal odometer is a device that measures distances in decimal units, such as kilometers or miles. It is based on the decimal number system, where each digit in the reading represents a power of 10.

What are the different types of number systems that a computer deals with?

Computers deal with various number systems, including:

Binary number system (base 2)
Octal number system (base 8)
Decimal number system (base 10)
Hexadecimal number system (base 16)

What do you understand by Binary Number system?

The binary number system is a base-2 number system that uses only two digits: 0 and 1. It is the most basic language that computers understand and is used to represent information in computers.

Explain Decimal Number system with an example.

The decimal number system is a base-10 number system that uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, the decimal number 456 can be represented as:

4 × 10^2 + 5 × 10^1 + 6 × 10^0 = 400 + 50 + 6 = 456

How will you convert:

(a) A decimal number to a binary number

To convert a decimal number to a binary number, we can divide the decimal number by 2 repeatedly and record the remainders. For example, to convert the decimal number 12 to binary:

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, we get the binary representation of 12: 1100

(b) A binary number to a decimal number

To convert a binary number to a decimal number, we can multiply each digit of the binary number by the corresponding power of 2 and add the results. For example, to convert the binary number 1100 to decimal:

1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 0 × 2^0 = 8 + 4 + 0 + 0 = 12

To convert an octal number to a binary number, we can convert each digit of the octal number to its binary equivalent and combine the results. For example, to convert the octal number 12 to binary:

1 = 001
2 = 010

Combining the binary equivalents, we get: 001010

(d) A binary number to a Hexa-decimal number

To convert a binary number to a hexadecimal number, we can divide the binary number into groups of 4 digits and convert each group to its hexadecimal equivalent. For example, to convert the binary number 11010110 to hexadecimal:

1101 = D
0110 = 6

Combining the hexadecimal equivalents, we get: D6

(e) A binary number to an octal number

To convert a binary number to an octal number, we can divide the binary number into groups of 3 digits and convert each group to its octal equivalent. For example, to convert the binary number 11010110 to octal:

110 = 6
101 = 5
110 = 6

Combining the octal equivalents, we get: 656

(f) A hexa-decimal number to an octal number

To convert a hexadecimal number to an octal number, we can convert the hexadecimal number to its binary equivalent and then convert the binary number to octal. For example, to convert the hexadecimal number 1A to octal:

1A = 00011010 (binary)

Dividing the binary number into groups of 3 digits, we get:

000 = 0
110 = 6
100 = 4

Combining the octal equivalents, we get: 064

What are the rules to perform:

(a) Binary Addition

Binary addition is performed by adding the corresponding digits of the two binary numbers, using the following rules:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1)

(b) Binary Subtraction

Binary subtraction is performed by subtracting the corresponding digits of the two binary numbers, using the following rules:

0 – 0 = 0
0 – 1 = 1 (borrow 1)
1 – 0 = 1
1 – 1 = 0

Write down the rules for:

(a) Binary Multiplication

Binary multiplication is performed by multiplying the corresponding digits of the two binary numbers, using the following rules:

0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1

(b) Binary Division

Binary division is performed by dividing the dividend by the divisor, using the following rules:

0 ÷ 0 = undefined
0 ÷ 1 = 0
1 ÷ 0 = undefined
1 ÷ 1 = 1

What do you understand by

(a) An Octal number

An octal number is a number that uses 8 digits: 0, 1, 2, 3, 4, 5, 6, and 7. It is a base-8 number system.

(b) A Hexa-decimal number

A hexadecimal number is a number that uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. It is a base-16 number system.

Give two differences between:

(a) Binary number and Decimal number

Binary numbers use only 2 digits (0 and 1), while decimal numbers use 10 digits (0-9).
Binary numbers are used by computers, while decimal numbers are used by humans.

(b) Octal number and Binary number

Octal numbers use 8 digits (0-7), while binary numbers use only 2 digits (0 and 1).
Octal numbers are often used as a shorthand for binary numbers.

Hexadecimal numbers use 16 digits (0-9, A-F), while octal numbers use 8 digits (0-7).
Hexadecimal numbers are often used to represent binary data in a more compact form.

Name the different types of operations that can be performed in Binary arithmetics

Addition
Subtraction
Multiplication
Division

12. Complete the following in tabular form

i) Octal Digit to Binary Equivalent

Octal Digit	Binary Equivalent
5	101
7	111
1	001
6	110
3	011

ii) Hexa-decimal Digit to Binary Equivalent

Hexadecimal Digit	Binary Equivalent
8	1000
11 (B)	1011
4	0100
15 (F)	1111
9	1001