Number Systems And Binary

Question 1

What are the two binary digits?

Answer 1

0 and 1

Question 2

What is the largest binary number that can be stored in 8 bits?

Answer 2

255 in decimal (binary: 11111111).

Question 3

What is the binary representation for the decimal number 134?

Answer 3

10000110 (128 + 4 + 2).

Question 4

Convert the binary number 011 to decimal.

Answer 4

3

Question 5

Convert the binary number 0110 to decimal.

Answer 5

6

Question 6

Convert the binary number 1001 to decimal.

Answer 6

9

Question 7

Convert the binary number 1010 to decimal.

Answer 7

10

Question 8

Convert the decimal number 5 to binary.

Answer 8

0101

Question 9

Convert the decimal number 7 to binary.

Answer 9

0111

Question 10

Convert the decimal number 26 to binary.

Answer 10

0011010

Question 11

What do binary fractions use after the binary point?

Answer 11

Positions (e.g., 0.5, 0.25).

Question 12

What is the binary representation for 136.75?

Answer 12

10001000.11 (128 + 8 + 0.5 + 0.25).

Question 13

What is a mixed number in binary?

Answer 13

Uses fixed bits for integers and fractions.

Question 14

What does 110011.01 equal in decimal?

Answer 14

51.25

Question 15

How to represent -5 in binary (using 8 bits)?

Answer 15

Convert 5 to binary: 00000101, flip the bits: 11111010, add 1: 11111011.

Question 16

How to represent -66 in binary (using 8 bits)?

Answer 16

Start with positive 66: 01000010, flip the bits: 10111101, add 1: 10111110.

Question 17

What are the addition rules in binary?

Answer 17

0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 0 (carry 1).

Question 18

What is an example of binary addition?

Answer 18

00011 + 00001 = 00100

Question 19

How to perform binary subtraction?

Answer 19

Convert second number to two’s complement and add.

Question 20

What is an example of binary subtraction?

Answer 20

12 - 9: 9: 00001001, Two’s complement: 11110111, Add to 12: 00001100 + 11110111 = 00000011 (3)

Question 21

What are the steps for binary multiplication?

Answer 21

Write binary numbers in a table, perform long multiplication (shift and add method).

Question 22

What is an example of binary multiplication?

Answer 22

8 (1000) × 3 (11) = 11000 (24).

Question 23

What is the binary multiplication of 8 × 2?

Answer 23

1000 × 10 = 10000 (16).

Question 24

What is the binary multiplication of 12 × 10?

Answer 24

1100 × 1010 = 1111000 (120).

Question 25

What is the repeated division-by-2 method?

Answer 25

A method to convert a decimal number to binary by dividing the number by 2 repeatedly and recording the remainders. The least significant bit (LSB) is at the top, and the most significant bit (MSB) is at the bottom.

Question 26

Convert the decimal number 294 to binary.

Answer 26

Using the division-by-2 method, 294 in binary is 100100110.

Question 27

Convert the decimal number 89 to binary.

Answer 27

Using the division-by-2 method, 89 in binary is 1011001.

Question 28

What is hexadecimal?

Answer 28

Hexadecimal is a base-16 numbering system using digits 0-9 and letters A-F. It is often used in computing for its compact representation of binary values.

Question 29

How do you convert binary to hexadecimal?

Answer 29

Group the binary number into 4-bit sections from right to left and convert each group into its hexadecimal equivalent.

Question 30

Convert the binary number 10010010 to hexadecimal.

Answer 30

Grouping into 4-bit sections: **1001 0010 The hexadecimal equivalent is **92

Question 31

What are the benefits of hexadecimal?

Answer 31

• Easier to read and understand than long binary strings.
• Compact representation of binary data.
• Reduces the chance of errors when typing or interpreting.

Question 32

Convert the hexadecimal number 3CF to binary.

Answer 32

Convert each hexadecimal digit: 3 = 0011, C = 1100, F = 1111. The binary equivalent is 001111001111.

Question 33

What are the components of floating-point numbers?

Answer 33

1. Significant: The main number digits.
2. Exponent: Determines the position of the decimal or binary point relative to the significant digits.

Question 34

Why are floating-point numbers needed?

Answer 34

They allow for the representation of very large or very small numbers in a compact format, suitable for scientific calculations and data storage.

Question 35

What is absolute error?

Answer 35

The difference between the estimated value and the actual value. For example, if the actual value is 20 and the estimated value is 18, the absolute error is 2.

Question 36

What is relative error?

Answer 36

The ratio of the absolute error to the actual value. For example, if the actual value is 20 and the absolute error is 2, the relative error is 2/20 = 0.1 or 10%.

Question 37

What are the limitations of floating-point numbers?

Answer 37

• Limited precision, leading to rounding errors.
• Cannot exactly represent some numbers.
• Susceptible to overflow and underflow in calculations.

Question 38

What are natural numbers and integers?

Answer 38

• Natural Numbers (N): Whole numbers starting from 0 (e.g., 0, 1, 2…).
• Integers (Z): Whole numbers including negatives (e.g., -3, -2, -1, 0, 1, 2…).

Question 39

What is the difference between rational and irrational numbers?

Answer 39

• Rational Numbers (Q): Numbers that can be expressed as a fraction (e.g., 7/1).
• Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2).

Question 40

What is a real number?

Answer 40

Rational and irrational numbers as a group

Question 41

What is a rounding error?

Answer 41

Decimal converted to floating point but can’t be represented exactly in available number of bits.