Number Conversions

Absolutely! Here’s a clear explanation that first covers the logic behind each type of conversion, then follows with examples. It’s simple, step-by-step, and easy to understand:

Number Conversions: Logic and Examples

1. Converting Binary to Denary (Decimal)

Logic:

Each digit in a binary number represents a power of 2, starting from at the rightmost bit.

To find the decimal value, multiply each binary digit by its corresponding power of 2.

Add up all the values where the binary digit is 1.

Example: Convert 11101110 to decimal

Place Value (powers of 2)	128	64	32	16	8	4	2	1
Binary Digit	1	1	1	0	1	1	1	0

Add the place values with 1s:
128 + 64 + 32 + 8 + 4 + 2 = 238

2. Converting Denary to Binary

Logic:

Divide the decimal number by 2 repeatedly.

At each division, note the remainder (0 or 1).

Continue until the quotient is 0.

The binary number is the remainders read from bottom to top.

Example: Convert 142 to binary

Divide by 2	Quotient	Remainder
142 ÷ 2	71	0
71 ÷ 2	35	1
35 ÷ 2	17	1
17 ÷ 2	8	1
8 ÷ 2	4	0
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Read remainders bottom to top → 10001110

3. Converting Hexadecimal to Binary

Logic:

Each hex digit represents a value from 0 to 15.

Convert each hex digit to its 4-bit binary equivalent.

Join all these binary groups together for the full binary number.

Example: Convert 21FD (hex) to binary

Hex Digit	2	1	F	D
Decimal	2	1	15	13
Binary	0010	0001	1111	1101

Final binary: 0010000111111101

4. Converting Binary to Hexadecimal

Logic:

Split the binary number into groups of 4 bits, starting from the right.

Add zeros on the left if the first group has less than 4 bits.

Convert each 4-bit group to decimal, then to hex (0–9 or A–F).

Example: Convert 1000011111101 to hex

Binary Group	0010	0001	1111	1101
Decimal	2	1	15	13
Hex	2	1	F	D

Final hex: 21FD

5. Converting Hexadecimal to Denary

Logic:

Convert hex to binary (as in step 3).

Convert that binary number to decimal (as in step 1).

6. Converting Denary to Hexadecimal

Logic:

Convert decimal to binary (as in step 2).

Convert binary to hex (as in step 4).

Sure! Here are detailed step-by-step solutions for each practice question:

Practice Questions & Worked Examples

1. Binary to Denary (Decimal)

Question: Convert binary 10110101 to decimal.

Step 1: Write place values (powers of 2):

128	64	32	16	8	4	2	1
1	0	1	1	0	1	0	1

Step 2: Add place values where bit = 1:

128 + 32 + 16 + 4 + 1 = 181

2. Denary to Binary

Question: Convert decimal 78 to binary.

Step 1: Divide by 2 repeatedly, write remainders:

Divide by 2	Quotient	Remainder
78 ÷ 2	39	0
39 ÷ 2	19	1
19 ÷ 2	9	1
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Step 2: Read remainders from bottom to top → 1001110

3. Hexadecimal to Binary

Question: Convert hex 3A9F to binary.

Hex Digit	3	A	9	F
Decimal	3	10	9	15
Binary	0011	1010	1001	1111

Final binary: 0011101010011111

4. Binary to Hexadecimal

Question: Convert binary 11011100101 to hexadecimal.

Step 1: Group bits into 4 from right (add leading zeros if needed):

Binary: 0001 1011 1001 0101

Group	0001	1011	1001	0101
Decimal	1	11	9	5
Hex	1	B	9	5

Final hex: 1B95

5. Hexadecimal to Denary

Question: Convert hex 4B7 to decimal.

Step 1: Convert each hex digit to binary:

Hex Digit	4	B	7
Decimal	4	11	7
Binary	0100	1011	0111

Binary number: 010010110111

Step 2: Convert binary to decimal:

Place Value	2048	1024	512	256	128	64	32	16	8	4	2	1
Binary Digit	0	1	0	0	1	0	1	1	0	1	1	1

Add values with 1: 1024 + 128 + 32 + 16 + 4 + 2 + 1 = 1207

6. Denary to Hexadecimal

Question: Convert decimal 255 to hexadecimal.

Step 1: Convert 255 to binary:

Divide by 2	Quotient	Remainder
255 ÷ 2	127	1
127 ÷ 2	63	1
63 ÷ 2	31	1
31 ÷ 2	15	1
15 ÷ 2	7	1
7 ÷ 2	3	1
3 ÷ 2	1	1
1 ÷ 2	0	1

Binary: 11111111

Step 2: Group binary into 4 bits:

1111 1111

Group	1111	1111
Dec	15	15
Hex	F	F

Final hex: FF