SSS 1 Mathematics Week 3
Topic: Addition, Subtraction, Multiplication and Division of Number Bases
Let’s first take a look at decimal addition.
As an example we have 26 plus 36,
26
+36
To add these two numbers, we first consider the “ones” column and calculate 6 plus 6, which results in 12. Since 12 is greater than 9 (remembering that base 10 operates with digits 0-9), we “carry” the 1 from the “ones” column to the “tens column” and leave the 2 in the “ones” column.
Considering the “tens” column, we calculate 1 + (2 + 3), which results in 6. Since 6 is less than 9, there is nothing to “carry” and we leave 6 in the “tens” column.
26
+36
62
Binary addition
Works in the same way, except that only 0’s and 1’s can be used, instead of the whole spectrum of 0-9. This actually makes binary addition much simpler than decimal addition, as we only need to remember the following:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10
As an example of binary addition we have,
101
+101
a) To add these two numbers, we first consider the “ones” column and calculate 1 + 1, which (in binary) results in 10. We “carry” the 1 to the “tens” column, and the leave the 0 in the “ones” column.
b) Moving on to the “tens” column, we calculate 1 + (0 + 0), which gives 1. Nothing “carries” to the “hundreds” column, and we leave the 1 in the “tens” column.
c) Moving on to the “hundreds” column, we calculate 1 + 1, which gives 10. We “carry” the 1 to the “thousands” column, leaving the 0 in the “hundreds” column.
101
+101
1010
Another example of binary addition:
1011
+1011
10110
Note that in the “tens” column, we have 1 + (1 + 1), where the first 1 is “carried” from the “ones” column. Recall that in binary,
1 + 1 + 1 = 10 + 1
= 11
Binary subtraction
is simplified as well, as long as we remember how subtraction and the base 2 number system. Let’s first look at an easy example.
111
– 10
101
Note that the difference is the same if this was decimal subtraction. Also similar to decimal subtraction is the concept of “borrowing.” Watch as “borrowing” occurs when a larger digit, say 8, is subtracted from a smaller digit, say 5, as shown below in decimal subtraction.
35
– 8
27
For 10 minus 1, 1 is borrowed from the “tens” column for use in the “ones” column, leaving the “tens” column with only 2. The following examples show “borrowing” in binary subtraction.
10 100 1010
– 1 – 10 – 110
1 10 100
Binary multiplication
Is actually much simpler than decimal multiplication. In the case of decimal multiplication, we need to remember 3 x 9 = 27, 7 x 8 = 56, and so on. In binary multiplication, we only need to remember the following,
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Note that since binary operates in base 2, the multiplication rules we need to remember are those that involve 0 and 1 only. As an example of binary multiplication we have 101 times 11,
101
x11
First we multiply 101 by 1, which produces 101. Then we put a 0 as a placeholder as we would in decimal multiplication, and multiply 101 by 1, which produces 101.
101
x11
101
1010 <– the 0 here is the placeholder
The next step, as with decimal multiplication, is to add. The results from our previous step indicates that we must add 101 and 1010, the sum of which is 1111.
101
x11
101
1010
1111
Binary division
Is almost as easy, and involves our knowledge of binary multiplication. Take for example the division of 1011 into 11.
11 R=10
11 )1011
-11
101
-11
10 <– remainder, R
To check our answer, we first multiply our divisor 11 by our quotient 11. Then we add its’ product to the remainder 10, and compare it to our dividend of 1011.
11
x 11
11
11
1001 <– product of 11 and 11
1001
+ 10
1011 <– sum of product and remainder
The sum is equal to our initial dividend, therefore our solution is correct.
Binary Numbers – Application to computer Programming
The binary numbers are numbers written in base 2, they have only two digits 0 and 1 which are very important in this computer age, the computer machine which is important today makes an extensive use of the off – on process which translates to mean the 0, 1 digits respectively, the binary numbers are used to express responses to statements or questions which have only two possible responses – “Yes” and “No”. Students who play game can make extensive use of the binary numbers. Having learnt that the binary numbers are made up of the two digits 0 and 1, they can code their messages and give responses to their messages.
Example: the answers to the following questions can be coded as follows:
Yes No
Have you paid your school fee? 1 0
Is 59 = 20 x 52 in indices 1 0
Is Log10 100 = 2? 1 0
Basically, the binary representation of a number is the way of writing that number, uniquely, using only the digits 0 and 1. These are the binary representations of the first 16 natural numbers:
Letters Numerical Codes Binary Equivalent/Codes
A 01 00001
B 02 00001
C 03 00010
D 04 00011
E 05 00100
F 06 00101
G 07 00110
H 08 00111
I 09 01000
J 10 01001
K 11 01010
L 12 01011
M 13 01100
N 14 01101
O 15 01110
P 16 01111
Q 17 10000
R 18 10001
S 19 10010
T 20 10011
U 21 10100
V 22 10101
W 23 10110
X 24 10111
Y 25 11000
Z 26 11010
Using the above binary codes, the statement below can be decoded, let us give it a trial!
“I am afraid of mathematics”
I a m a f r a i d o f m a t
01001, 00001 01101, 00001 00110 10010 00001 01001 00100, 01111 00110, 01101 00001 10100
h e m a t i c s
01000 00101 01101 00001 10100 01001 00011 10011
Note that it is always good to use the alternative method, i.e. conversion to denary then to binary because the digital conversion method can be confusing at times if care is not taken.
Exercises:
Solve these additions of number bases
1. 36 + 46
A. 81 B. 82 C. 83 D. 84
2. 101 + 11
A.1001 B. 1010 C. 1000 D. 1011
Binary subtraction
3. 110 – 10
A.101 B. 110 C. 110 D. 100
4. 100 x 11
A. 1000 B. 1100 C. 1001 D. 1101
Represent this statement in binary codes:
5. “I do not like Harrison Stone”
Answers:
1. B 2. C 3. D 4. B
3 thoughts on “Classwork Series and Exercises {Mathematics – SS1}: Addition, Subtraction, Multiplication and Division of Number Bases”
very helpful and informative, thanks very much.
Its educative and impartive tuitor
Good