Question

convert the following real numbers to binary (five binary places)  a.27.15

Answer 1

To convert the decimal number 27.15 into binary with five binary places, we need to convert both the integer part (27) and the fractional part (0.15) separately.

**Step 1: Convert the integer part (27) to binary.**

- Divide the number by 2 and record the remainder.
- Continue dividing the quotient until it is 0.

```
27 ÷ 2 = 13 remainder 1
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
```

Now, read the remainders from bottom to top: **11011**

**Step 2: Convert the fractional part (0.15) to binary.**

- Multiply the fractional part by 2 and take the integer part as the next binary digit.
- Continue with the new fractional part until you reach five binary places.

```
0.15 * 2 = 0.30 → Integer part: 0
0.30 * 2 = 0.60 → Integer part: 0
0.60 * 2 = 1.20 → Integer part: 1
0.20 * 2 = 0.40 → Integer part: 0
0.40 * 2 = 0.80 → Integer part: 0
```

Now, the integer parts obtained are: **00100**

**Step 3: Combine the integer and fractional parts.**

The binary representation of 27.15 is:

```
27.15 (Decimal) = 11011.00100 (Binary)
```

So, in binary with five binary places for the fraction, **27.15** is represented as **11011.00100**.