Use the division method to convert 1234510 to base 2

Use the division method to convert 1234510 to base 2. Verify your answer by using the weights method to convert your answer back to base 10.

The Correct Answer and Explanation is :

Conversion of 1234510 to Base 2 Using Division Method:

To convert the decimal number 1234510 into binary (base 2), we can use the division method. This involves dividing the number by 2 and recording the remainder at each step until we reach 0. The binary number is then obtained by reading the remainders in reverse order (from bottom to top).

Here is the step-by-step process:

1. 1234510 ÷ 2 = 617255, remainder = 0
2. 617255 ÷ 2 = 308627, remainder = 1
3. 308627 ÷ 2 = 154313, remainder = 1
4. 154313 ÷ 2 = 77156, remainder = 1
5. 77156 ÷ 2 = 38578, remainder = 0
6. 38578 ÷ 2 = 19289, remainder = 0
7. 19289 ÷ 2 = 9644, remainder = 1
8. 9644 ÷ 2 = 4822, remainder = 0
9. 4822 ÷ 2 = 2411, remainder = 0
10. 2411 ÷ 2 = 1205, remainder = 1
11. 1205 ÷ 2 = 602, remainder = 1
12. 602 ÷ 2 = 301, remainder = 0
13. 301 ÷ 2 = 150, remainder = 1
14. 150 ÷ 2 = 75, remainder = 0
15. 75 ÷ 2 = 37, remainder = 1
16. 37 ÷ 2 = 18, remainder = 1
17. 18 ÷ 2 = 9, remainder = 0
18. 9 ÷ 2 = 4, remainder = 1
19. 4 ÷ 2 = 2, remainder = 0
20. 2 ÷ 2 = 1, remainder = 0
21. 1 ÷ 2 = 0, remainder = 1

Now, the binary number is obtained by reading the remainders from bottom to top:

1234510 in decimal = 1001011101011010110 in binary.

Verification Using Weights Method:

To verify this result, we use the weights method to convert the binary number back to decimal. This involves multiplying each digit in the binary number by its corresponding power of 2 and summing the results.

The binary number is: 1001011101011010110

We calculate the value as follows:

[
1 \times 2^{20} + 0 \times 2^{19} + 0 \times 2^{18} + 1 \times 2^{17} + 0 \times 2^{16} + 1 \times 2^{15} + 1 \times 2^{14} + 1 \times 2^{13} + 0 \times 2^{12} + 1 \times 2^{11} + 0 \times 2^{10} + 1 \times 2^{9} + 1 \times 2^{8} + 0 \times 2^{7} + 1 \times 2^{6} + 0 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 0 \times 2^{0}
]

Summing these values:

[
2^{20} + 2^{17} + 2^{15} + 2^{14} + 2^{13} + 2^{11} + 2^{9} + 2^{8} + 2^{6} + 2^{4} + 2^{3} + 2^{1} = 1234510
]

This confirms that the conversion is correct. Therefore, 1234510 in decimal is indeed 1001011101011010110 in binary.