Number System and Base Conversion

Any base to Base10, Base10 to any Base, Any Other Base to any Other Base

1. From any base to Decimal (Base-10)

• Multiply each digit by its base raised to the positional power.
• Add all results.
• Example: (1011)₂ to decimal:

1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8 + 0 + 2 + 1
= 11₁₀

If the binary number is a float, you convert integer part and fractional part separately: ⭐

Steps:

1. Integer part → Multiply each bit by 2^n according to its position (same as normal).
2. Fractional part → Multiply each bit by 2^(-n) according to its position after the point.

• Example: (11001.101)₂ to base-10

Integer part (11001):

1×2⁴ + 1×2³ + 0×2² + 0×2¹ + 1×2⁰
= 16 + 8 + 0 + 0 + 1
= 25

Fractional part (.101):

1×2⁻¹ + 0×2⁻² + 1×2⁻³
= 0.5 + 0 + 0.125
= 0.625

Final:
(11001.101)₂ = (25.625)₁₀

2. From Decimal to Any Base

• Repeatedly divide the decimal number by the target base.
• Write remainders in reverse order.
• Example: (25)₁₀ to base-2:

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
→ 11001₂

If the decimal number is a float, you convert integer part and fractional part separately: ⭐

Steps:

1. Integer part → Use repeated division by base (same as normal).
2. Fractional part → Multiply repeatedly by base, take the integer part each time as the next digit.

• Example: (25.625)₁₀ to base-2

Integer part (25):

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
→ 11001

Fractional part (0.625):

0.625 × 2 = 1.25   → 1
0.25  × 2 = 0.5    → 0
0.5   × 2 = 1.0    → 1
→ .101

Final:
(25.625)₁₀ = (11001.101)₂

3. From One Non-Decimal Base to Another

• Convert to decimal first, then to target base.
• Example: (27)₈ to base-2:

1. (27)₈ = 2×8¹ + 7×8⁰
         = 16 + 7
         = 23₁₀

2. 23₁₀ = 10111₂

---

Base Conversion

1. Binary to Decimal Conversion

Steps:

1. Write Down the Binary Number:
   • Example: 1101
2. Identify the Positional Values:
   • Each bit represents a power of 2, starting from (2^0) for the rightmost bit.
3. Multiply Each Bit by Its Corresponding Power of 2:
   • For 1101:
     • 1 \times 2^3 = 8
     • 1 \times 2^2 = 4
     • 0 \times 2^1 = 0
     • 1 \times 2^0 = 1
4. Sum the Results:
   • 8 + 4 + 0 + 1 = 13

Result:

• Binary 1101 is decimal 13.

2. Decimal to Binary Conversion

Steps:

1. Divide the Decimal Number by 2:
   • Record the remainder.
2. Divide the Quotient by 2:
   • Continue dividing and recording remainders until the quotient is 0.
3. Write Down the Remainders in Reverse Order:
   • This gives the binary representation.

Example: Convert 13 to Binary

• 13 ÷ 2 = 6 with remainder 1
• 6 ÷ 2 = 3 with remainder 0
• 3 ÷ 2 = 1 with remainder 1
• 1 ÷ 2 = 0 with remainder 1

Result:

• Binary representation is 1101.

3. Hexadecimal to Decimal Conversion

Steps:

1. Write Down the Hexadecimal Number:
   • Example: 0x2F3
2. Identify the Positional Values:
   • Each digit represents a power of 16, starting from (16^0) for the rightmost digit.
3. Convert Each Hexadecimal Digit to Decimal:
   • 2, F (15), and 3.
4. Multiply Each Digit by Its Corresponding Power of 16:
   • For 0x2F3:
     • 3 \times 16^0 = 3
     • 15 \times 16^1 = 240
     • 2 \times 16^2 = 512
5. Sum the Results:
   • 512 + 240 + 3 = 755

Result:

• Hexadecimal 0x2F3 is decimal 755.

4. Decimal to Hexadecimal Conversion

Steps:

1. Divide the Decimal Number by 16:
   • Record the remainder (convert to hexadecimal if needed).
2. Divide the Quotient by 16:
   • Continue dividing and recording remainders until the quotient is 0.
3. Write Down the Remainders in Reverse Order:
   • This gives the hexadecimal representation.

Example: Convert 755 to Hexadecimal

• 755 ÷ 16 = 47 with remainder 7
• 47 ÷ 16 = 2 with remainder 15 (F in hexadecimal)
• 2 ÷ 16 = 0 with remainder 2

Result:

• Hexadecimal representation is 0x2F7.

5. Binary to Hexadecimal Conversion

Steps:

1. Group the Binary Digits into Sets of Four (from right to left):
   • Example: 10111011 becomes 1011 1011.
2. Convert Each Group to Hexadecimal:
   • 1011 is B
   • 1011 is B

Result:

• Binary 10111011 is hexadecimal 0xBB.

6. Hexadecimal to Binary Conversion

Steps:

1. Convert Each Hexadecimal Digit to Its 4-Bit Binary Equivalent:
   • Example: 0x2F3:
     • 2 is 0010
     • F is 1111
     • 3 is 0011
2. Combine All Binary Groups:

Result:

• Hexadecimal 0x2F3 is binary 0010 1111 0011.

---