Question

1. add the following 8-bit binary numbers .

i. (01001110)2 + (00111100)2

ii.(10011101)2 + (10001111)2

Answer 1

Answer on Question #57255 - Math - Discrete Math

Question

1. Add the following 8-bit binary numbers.

i. $(01001110)2 + (00111100)2$

ii. $(10011101)2 + (10001111)2$

Solution

i.

We start with the two numbers in the far right column, add the numbers following the rules of addition

$$0 + 0 = 0, \quad 0 + 1 = 1, \quad 1 + 0 = 1,$$

Unless both numbers are 1.

```

0 1 0 0 1 1 1 0

0 0 1 1 1 1 0 0

```

If both numbers are 1, then

$$1 + 1 = 10$$

and write 0 below and carry '1' to the next column.

Start on the next column to the left hand side

0 1 0 0 1 1 1 0

0 0 1 1 1 1 0 0

1 0

Repeat the steps above, but add any carry. Remember that 1+1=10 and 1+1+1=11

0 1 0 0 1 1 1 0

0 0 1 1 1 1 0 0

1 0 0 0 1 0 1 0

Thus,

$$(01001110)_2 + (010111100)_2 = (10001010)_2$$

ii.

We start with the two numbers in the far right column, add the numbers following the rules of addition

$$0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1,$$

Unless both numbers are 1.

If both numbers are 1, then

$$1 + 1 = 10$$

and write 0 below and carry '1' to the next column.

1 0 0 1 1 1 0 1

1 0 0 0 1 1 1 1

Start on the next column to the left hand side. Repeat the steps above, but add any carry. Rememeber that 1+1=10 and 1+1+1=11

1 0 0 1 1 1 0 1

1 0 0 0 1 1 1 1

0 0

Repeat the steps above, but add any carry. Rememeber that 1+1=10 and 1+1+1=11

1 0 0 1 1 1 0 1

1 0 0 0 1 1 1 1

1 0 0 1 0 1 1 0 0

Thus,

$$(10011101)_2 + (10001111)_2 = (100101100)_2 - \text{overflow}.$$

Answer: i. $(10001010)_2$. ii. $(100101100)_2$.

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