# 32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 12.342 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

## Number 12.342(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

### 1. First, convert to binary (in base 2) the integer part: 12. Divide the number repeatedly by 2.

#### We stop when we get a quotient that is equal to zero.

• division = quotient + remainder;
• 12 ÷ 2 = 6 + 0;
• 6 ÷ 2 = 3 + 0;
• 3 ÷ 2 = 1 + 1;
• 1 ÷ 2 = 0 + 1;

### 3. Convert to binary (base 2) the fractional part: 0.342.

#### Stop when we get a fractional part that is equal to zero.

• #) multiplying = integer + fractional part;
• 1) 0.342 × 2 = 0 + 0.684;
• 2) 0.684 × 2 = 1 + 0.368;
• 3) 0.368 × 2 = 0 + 0.736;
• 4) 0.736 × 2 = 1 + 0.472;
• 5) 0.472 × 2 = 0 + 0.944;
• 6) 0.944 × 2 = 1 + 0.888;
• 7) 0.888 × 2 = 1 + 0.776;
• 8) 0.776 × 2 = 1 + 0.552;
• 9) 0.552 × 2 = 1 + 0.104;
• 10) 0.104 × 2 = 0 + 0.208;
• 11) 0.208 × 2 = 0 + 0.416;
• 12) 0.416 × 2 = 0 + 0.832;
• 13) 0.832 × 2 = 1 + 0.664;
• 14) 0.664 × 2 = 1 + 0.328;
• 15) 0.328 × 2 = 0 + 0.656;
• 16) 0.656 × 2 = 1 + 0.312;
• 17) 0.312 × 2 = 0 + 0.624;
• 18) 0.624 × 2 = 1 + 0.248;
• 19) 0.248 × 2 = 0 + 0.496;
• 20) 0.496 × 2 = 0 + 0.992;
• 21) 0.992 × 2 = 1 + 0.984;
• 22) 0.984 × 2 = 1 + 0.968;
• 23) 0.968 × 2 = 1 + 0.936;
• 24) 0.936 × 2 = 1 + 0.872;

### 9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

#### Use the same technique of repeatedly dividing by 2:

• division = quotient + remainder;
• 130 ÷ 2 = 65 + 0;
• 65 ÷ 2 = 32 + 1;
• 32 ÷ 2 = 16 + 0;
• 16 ÷ 2 = 8 + 0;
• 8 ÷ 2 = 4 + 0;
• 4 ÷ 2 = 2 + 0;
• 2 ÷ 2 = 1 + 0;
• 1 ÷ 2 = 0 + 1;