Question

1. Perform the following arithmetic (show your work):

Base 2

01101101                   10010110

+

00001111                  00100011

Base 16

ABCD                          1234

+                               -

1234                             ABC

Answer 1

Binary addition is similar to normal addition except that two is written as 10 in binary after addition.

After summing 1 + 1 in binary, we get 2 but we represent it like 10 by a digit 0 and a carry of 1.

1)

01101101

+

00001111

--------------------

01111100

--------------------

The above sum is carried out by following steps[from right] -

1+1 = 0 with a carry of 1

1+0+1=0 with a carry of 1

1+1+1 =1 with a carry of 1

1+1+1 =1 with a carry of 1

1+0+0 = 1

1+0=1

1+0=1

0+0=0

Hence result of addition = 01111100

2)

10010110

+

00100011

--------------------

10111001

--------------------

The above sum is carried out by following steps[from right] -

0+1 = 1

1+1=0 with a carry of 1

1+1+0 =0 with a carry of 1

1+0+0 =1

1+0 = 1

0+1=1

0+0=0

1+0=1

Hence result of addition = 10111001

Subtraction in binary/base2

10010110

-

00100011

--------------------

01110011

--------------------

We can do subtraction in base 2 using complement Method as shown below

step 1 - Represent both the numbers in same number of digits . Here both 10010110 and 00100011 have 8 digits .but if any one number had say 7 digits [0100011] then we add zeros in the start i.e. 00100011

step 2 - change all 0s to 1s and 1s to 0s in second term . Here first term- 10010110 and Second term -00100011

So we change 00100011 to 11011100 . And add 1 to the newly obtained term 11011100+1 =11011101

step 3 - We add newly obtained term with first term

11011101

+

10010110

--------------------

101110011

--------------------

step 4 - Discard the first digit , so after discarding first digit i.e. 1 , we have 01110011

Hence result of addition = 01110011

3)

Decimal to Hex converison table -

Decimal base 10	Hex base 16
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	A
11	B
12	C
13	D
14	E
15	F
16	10
17	11
18	12
19	13
20	14
21	15
22	16
23	17
24	18
25	19
26	1A
27	1B
28	1C
29	1D
30	1E
40	28
50	32
60	3C
70	46
80	50
90	5A
100	64
200	C8
1000	3E8
2000	7D0

ABCD

+

1234

-------------

BE01

-------------

Explanation:

1. D+4= 13(as per hex conversion table above) +4 = 17

Now 17 is not less than 16 , so we subtract 17-16 = 1 . And we add a carry of 1 to the left .

Note : if it was less than 16 then we dont do above operation of subtracting 16 and then adding a carry of 1 to left.

2. C+3+1(carry) = 12+3+1 = 16

So we subtract 16-16 = 0. And we add a carry of 1 to the left .

3. B+2 +1(carry) = 11+3=14 = E(as per hex table above).

4. A+1 = 10 +1 = 11 = B(as per hex table above)

Hence result of addition = BE01

4)

Hexadecimal Subtraction:

1234

-

ABC

-------------

778

-------------

In the decimal system, you borrow 10, but in hexadecimal system you borrow 16.

i. 4-C = 4-12(as per hex table) = -8 ,since result is negative we borrow 16 from 3 .

so 4-12+16=8

ii. 3-B-1(borrow) = 3-11-1 = -9 ,since result is negative we borrow 16 from 2 .

so 3-11-1+16 = 7

iii. 2-A-1(borrow) = 2-10-1 =-9 , since result is negative we borrow 16 from 1 .

so 2-10-1+16 = 7

iv. 1-1(borrow) = 0

Hence result of subtraction = 778