Question
convert the following real numbers to binary (five binary places) a.43.75
Answers
GPT-4o mini
To convert the decimal number 43.75 to binary with five binary places, we need to convert both the integer part (43) and the fractional part (0.75) separately.
### Step 1: Convert the integer part (43) to binary
1. Divide the integer by 2 and record the remainder.
2. Continue dividing the quotient by 2 until it equals 0.
3. The binary representation is the remainders read in reverse order.
Let's do this for 43:
- \( 43 \div 2 = 21 \) remainder **1**
- \( 21 \div 2 = 10 \) remainder **1**
- \( 10 \div 2 = 5 \) remainder **0**
- \( 5 \div 2 = 2 \) remainder **1**
- \( 2 \div 2 = 1 \) remainder **0**
- \( 1 \div 2 = 0 \) remainder **1**
Reading the remainders from bottom to top, we find that:
**43 in binary is** \(101011\).
### Step 2: Convert the fractional part (0.75) to binary
To convert the fractional part, we multiply by 2 and take the integer part as the binary digit. Then, we continue with the new fractional part.
1. \(0.75 \times 2 = 1.5\) (the integer part is **1**)
2. \(0.5 \times 2 = 1.0\) (the integer part is **1**)
3. \(0.0\) (stop here since the fractional part is now 0)
So, the binary representation of the fractional part (0.75) is:
**0.75 in binary is** \(0.11\).
### Combine both parts
Now we combine the integer and fractional parts:
**43.75 in binary is** \(101011.11\).
### Format for five binary places
We need to represent the fractional part with 5 binary places. Since \(0.75\) is \(0.11\) in binary (which uses 2 places), we will pad with additional zeros to make it 5 places:
**Final Binary Representation:**
**43.75 in binary (with five binary places)** is:
\[
101011.11000
\]
### Step 1: Convert the integer part (43) to binary
1. Divide the integer by 2 and record the remainder.
2. Continue dividing the quotient by 2 until it equals 0.
3. The binary representation is the remainders read in reverse order.
Let's do this for 43:
- \( 43 \div 2 = 21 \) remainder **1**
- \( 21 \div 2 = 10 \) remainder **1**
- \( 10 \div 2 = 5 \) remainder **0**
- \( 5 \div 2 = 2 \) remainder **1**
- \( 2 \div 2 = 1 \) remainder **0**
- \( 1 \div 2 = 0 \) remainder **1**
Reading the remainders from bottom to top, we find that:
**43 in binary is** \(101011\).
### Step 2: Convert the fractional part (0.75) to binary
To convert the fractional part, we multiply by 2 and take the integer part as the binary digit. Then, we continue with the new fractional part.
1. \(0.75 \times 2 = 1.5\) (the integer part is **1**)
2. \(0.5 \times 2 = 1.0\) (the integer part is **1**)
3. \(0.0\) (stop here since the fractional part is now 0)
So, the binary representation of the fractional part (0.75) is:
**0.75 in binary is** \(0.11\).
### Combine both parts
Now we combine the integer and fractional parts:
**43.75 in binary is** \(101011.11\).
### Format for five binary places
We need to represent the fractional part with 5 binary places. Since \(0.75\) is \(0.11\) in binary (which uses 2 places), we will pad with additional zeros to make it 5 places:
**Final Binary Representation:**
**43.75 in binary (with five binary places)** is:
\[
101011.11000
\]