Answer :
To convert the given base 10 numbers to the specified bases, let's follow a step-by-step process for each conversion:
### I. Converting [tex]\(0.653_{\text{ten}}\)[/tex] to base two (binary)
1. Initialize a variable to store the binary fraction: Let's start with an empty string to build the binary representation of the fractional part.
2. Multiply the fractional part by 2: Take the decimal fractional part and keep multiplying it by 2. For each step:
- If the result is greater than or equal to 1, append `1` to the binary fraction string and subtract 1 from the result.
- Otherwise, append `0` to the binary fraction string.
Let's illustrate these steps:
- [tex]\(0.653 \times 2 = 1.306\)[/tex] (append `1`, new fraction part is 0.306)
- [tex]\(0.306 \times 2 = 0.612\)[/tex] (append `0`, new fraction part is 0.612)
- [tex]\(0.612 \times 2 = 1.224\)[/tex] (append `1`, new fraction part is 0.224)
- [tex]\(0.224 \times 2 = 0.448\)[/tex] (append `0`, new fraction part is 0.448)
- [tex]\(0.448 \times 2 = 0.896\)[/tex] (append `0`, new fraction part is 0.896)
- [tex]\(0.896 \times 2 = 1.792\)[/tex] (append `1`, new fraction part is 0.792)
- [tex]\(0.792 \times 2 = 1.584\)[/tex] (append `1`, new fraction part is 0.584)
- [tex]\(0.584 \times 2 = 1.168\)[/tex] (append `1`, new fraction part is 0.168)
- [tex]\(0.168 \times 2 = 0.336\)[/tex] (append `0`, new fraction part is 0.336)
- [tex]\(0.336 \times 2 = 0.672\)[/tex] (append `0`, new fraction part is 0.672)
- [tex]\(0.672 \times 2 = 1.344\)[/tex] (append `1`, new fraction part is 0.344)
- [tex]\(0.344 \times 2 = 0.688\)[/tex] (append `0`, new fraction part is 0.688)
- [tex]\(0.688 \times 2 = 1.376\)[/tex] (append `1`, new fraction part is 0.376)
- [tex]\(0.376 \times 2 = 0.752\)[/tex] (append `0`, new fraction part is 0.752)
- [tex]\(0.752 \times 2 = 1.504\)[/tex] (append `1`, new fraction part is 0.504)
- [tex]\(0.504 \times 2 = 1.008\)[/tex] (append `1`, new fraction part is 0.008)
- [tex]\(0.008 \times 2 = 0.016\)[/tex] (append `0`, new fraction part is 0.016)
- [tex]\(0.016 \times 2 = 0.032\)[/tex] (append `0`, new fraction part is 0.032)
- [tex]\(0.032 \times 2 = 0.064\)[/tex] (append `0`, new fraction part is 0.064)
- [tex]\(0.064 \times 2 = 0.128\)[/tex] (append `0`, new fraction part is 0.128)
After a sufficient number of steps for practicality (in this case, 20 steps), we get the binary fraction representation:
[tex]\[ 0.653_{\text{ten}} = 0.10100111001010110000_2 \][/tex]
### II. Converting [tex]\(578.38_{\text{ten}}\)[/tex] to base five
1. Convert the integer part (578) to base five:
- Divide 578 by 5: quotient is 115, remainder is 3.
- Divide 115 by 5: quotient is 23, remainder is 0.
- Divide 23 by 5: quotient is 4, remainder is 3.
- Divide 4 by 5: quotient is 0, remainder is 4.
Reading the remainders from bottom to top gives us the base five integer part:
[tex]\[ 578_{\text{ten}} = 4303_5 \][/tex]
2. Convert the fractional part (0.38) to base five:
Similar to the binary conversion, multiply the fractional part by 5 and track the integer parts:
- [tex]\(0.38 \times 5 = 1.90\)[/tex] (append `1`, new fraction part is 0.90)
- [tex]\(0.90 \times 5 = 4.50\)[/tex] (append `4`, new fraction part is 0.50)
- [tex]\(0.50 \times 5 = 2.50\)[/tex] (append `2`, new fraction part is 0.50)
- [tex]\(0.50 \times 5 = 2.50\)[/tex] (append `2`, new fraction part is 0.50)
- Continue similarly...
After sufficient steps, we get the base five fraction representation:
[tex]\[ 0.38_{\text{ten}} \approx 0.14222222222222222222_5 \][/tex]
Combining both parts, the conversion is:
[tex]\[ 578.38_{\text{ten}} = 4303.14222222222222222222_5 \][/tex]
### I. Converting [tex]\(0.653_{\text{ten}}\)[/tex] to base two (binary)
1. Initialize a variable to store the binary fraction: Let's start with an empty string to build the binary representation of the fractional part.
2. Multiply the fractional part by 2: Take the decimal fractional part and keep multiplying it by 2. For each step:
- If the result is greater than or equal to 1, append `1` to the binary fraction string and subtract 1 from the result.
- Otherwise, append `0` to the binary fraction string.
Let's illustrate these steps:
- [tex]\(0.653 \times 2 = 1.306\)[/tex] (append `1`, new fraction part is 0.306)
- [tex]\(0.306 \times 2 = 0.612\)[/tex] (append `0`, new fraction part is 0.612)
- [tex]\(0.612 \times 2 = 1.224\)[/tex] (append `1`, new fraction part is 0.224)
- [tex]\(0.224 \times 2 = 0.448\)[/tex] (append `0`, new fraction part is 0.448)
- [tex]\(0.448 \times 2 = 0.896\)[/tex] (append `0`, new fraction part is 0.896)
- [tex]\(0.896 \times 2 = 1.792\)[/tex] (append `1`, new fraction part is 0.792)
- [tex]\(0.792 \times 2 = 1.584\)[/tex] (append `1`, new fraction part is 0.584)
- [tex]\(0.584 \times 2 = 1.168\)[/tex] (append `1`, new fraction part is 0.168)
- [tex]\(0.168 \times 2 = 0.336\)[/tex] (append `0`, new fraction part is 0.336)
- [tex]\(0.336 \times 2 = 0.672\)[/tex] (append `0`, new fraction part is 0.672)
- [tex]\(0.672 \times 2 = 1.344\)[/tex] (append `1`, new fraction part is 0.344)
- [tex]\(0.344 \times 2 = 0.688\)[/tex] (append `0`, new fraction part is 0.688)
- [tex]\(0.688 \times 2 = 1.376\)[/tex] (append `1`, new fraction part is 0.376)
- [tex]\(0.376 \times 2 = 0.752\)[/tex] (append `0`, new fraction part is 0.752)
- [tex]\(0.752 \times 2 = 1.504\)[/tex] (append `1`, new fraction part is 0.504)
- [tex]\(0.504 \times 2 = 1.008\)[/tex] (append `1`, new fraction part is 0.008)
- [tex]\(0.008 \times 2 = 0.016\)[/tex] (append `0`, new fraction part is 0.016)
- [tex]\(0.016 \times 2 = 0.032\)[/tex] (append `0`, new fraction part is 0.032)
- [tex]\(0.032 \times 2 = 0.064\)[/tex] (append `0`, new fraction part is 0.064)
- [tex]\(0.064 \times 2 = 0.128\)[/tex] (append `0`, new fraction part is 0.128)
After a sufficient number of steps for practicality (in this case, 20 steps), we get the binary fraction representation:
[tex]\[ 0.653_{\text{ten}} = 0.10100111001010110000_2 \][/tex]
### II. Converting [tex]\(578.38_{\text{ten}}\)[/tex] to base five
1. Convert the integer part (578) to base five:
- Divide 578 by 5: quotient is 115, remainder is 3.
- Divide 115 by 5: quotient is 23, remainder is 0.
- Divide 23 by 5: quotient is 4, remainder is 3.
- Divide 4 by 5: quotient is 0, remainder is 4.
Reading the remainders from bottom to top gives us the base five integer part:
[tex]\[ 578_{\text{ten}} = 4303_5 \][/tex]
2. Convert the fractional part (0.38) to base five:
Similar to the binary conversion, multiply the fractional part by 5 and track the integer parts:
- [tex]\(0.38 \times 5 = 1.90\)[/tex] (append `1`, new fraction part is 0.90)
- [tex]\(0.90 \times 5 = 4.50\)[/tex] (append `4`, new fraction part is 0.50)
- [tex]\(0.50 \times 5 = 2.50\)[/tex] (append `2`, new fraction part is 0.50)
- [tex]\(0.50 \times 5 = 2.50\)[/tex] (append `2`, new fraction part is 0.50)
- Continue similarly...
After sufficient steps, we get the base five fraction representation:
[tex]\[ 0.38_{\text{ten}} \approx 0.14222222222222222222_5 \][/tex]
Combining both parts, the conversion is:
[tex]\[ 578.38_{\text{ten}} = 4303.14222222222222222222_5 \][/tex]
