Question

Convert the following decimal numbers into equivalent binary numbers and then convert the binary answer back into equivalent decimal. Show all the steps of conversions.

• -8735

Answer 1

`-8375_(10) = -10001000011111_2`

`-10001000011111_2 = -8735_{10`

Step-by-step explanation:

Given

-8735

Required

Convert to binary and then back to decimal

To convert to binary, we divide by 2 and write down the remainder (ignore the negative sign, first)

8735 / 2 = 4367 R 1

4367 / 2 = 2183 R 1

2183 / 2 = 1091 R 1

1091 / 2 = 545 R 1

545 / 2 = 272 R 1

272 / 2 = 136 R 0

136 / 2 = 68 R 0

68 / 2 = 34 R 0

34 / 2 = 17 R 0

17 / 2 = 8 R 1

8 / 2 = 4 R 0

4 / 2 = 2 R 0

2 / 2 = 1 R 0

1 / 2 = 0 R 1

Write out the remainder from bottom to top

`8735_(10) = 10001000011111_2`

Include the negative:

`-8735_(10) = -10001000011111_2`

To convert to decimal, we simply multiply the individual digits with the base (in this case, it is base 2) raise to power of the index of that digit.

There are 14 digits in 10001000011111 and the rightmost digit is of index 0 and it increased to the left.

Ignore the negative, first; so, we have:

`10001000011111_2 = 1 * 2^(13) + 0 * 2^(12) + 0 * 2^(11) + 0 * 2^(10) + 1 * 2^(9) + 0 * 2^(8) + 0 * 2^(7) + 0 * 2^(6) + 0 * 2^(5) + 1 * 2^(4) + 1 * 2^(3) + 1 * 2^(2) + 1* 2^(1) + 1 * 2^0`

`10001000011111_2 = 1 * 8192 + 0 + 0 + 0 + 1 * 512 + 0 + 0 + 0 + 0 + 1 * 16 + 1 * 8 + 1 * 4 + 1* 2 + 1 * 1`

`10001000011111_2 = 8192 + 512 + 16 + 8 + 4 + 2 + 1`

`10001000011111_2 = 8735`

Include the negative sign

`-10001000011111_2 = -8735_{10`