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Q35E

Expert-verified
Found in: Page 256

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# 35. What integer does each of the following one’s complement representations of length five represent? a) 11001 b) 01101 c) 10001 d) 11111

The integer have one’s complement representations of length five represent

$\left(a\right)-6\phantom{\rule{0ex}{0ex}}\left(b\right)13\phantom{\rule{0ex}{0ex}}\left(c\right)-14\phantom{\rule{0ex}{0ex}}\left(d\right)0$

See the step by step solution

## Step 1:

(a) $11001$

The one’s complement representation starts with a 1, which indicates that the number is negative

Let us remove the first digit:

$1001$

Then replace every 1 by 0 and every 0 by 1

$0110$

Determine the corresponding integer in decimal notation:

$\left(0110{\right)}_{2}={0.2}^{3}+{1.2}^{2}+1.2+0=4+2=6$

We also knew that the integer had to be negative, thus the one’s complement $11001$then corresponds with the integer $-6$

## Step 2:

(b) $01101$

The one’s complement representation starts with a 0, which indicates that the number is positive

Let us remove the first digit:

$1101$

Determine the corresponding integer in decimal notation:

$\left(1101{\right)}_{2}={1.2}^{3}+{1.2}^{2}+0.2+1=8+4+1=13$

We also knew that the integer had to be positive, thus the one’s complement $01101$then corresponds with the integer $13$

## Step 3:

$10001$

The one’s complement representation starts with a 1, which indicates that the number is negative

Let us remove the first digit:

$0001$

Then replace every 1 by 0 and every 0 by 1

$1110$

Determine the corresponding integer in decimal notation:

$\left(1101{\right)}_{2}={1.2}^{3}+{1.2}^{2}+1.2+1=8+4+2=14$

We also knew that the integer had to be negative, thus the one’s complement $10001$ then corresponds with the integer $-14$

## Step 4:

(c)$11111$

The one’s complement representation starts with a 1, which indicates that the number is negative

Let us remove the first digit:

$1111$

Then replace every 1 by 0 and every 0 by 1

$0000$

Determine the corresponding integer in decimal notation:

$\left(0000{\right)}_{2}={0.2}^{3}+{0.2}^{2}+0.2+0=0$

We also knew that the integer had to be negative, thus the one’s complement $11111$then corresponds with the integer $-0=0$