# Lesson Notes By Weeks and Term - Junior Secondary School 1

BASE NUMBERS

SUBJECT: MATHEMATICS

CLASS:  JSS 1

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

• New General Mathematics, Junior Secondary Schools Book 1
• Essential Mathematics for Junior Secondary Schools Book 1

WEEK FOUR                                    Date :……….

TOPIC: BASE NUMBERS

Content

• Number Bases ( Expansion  of Base Numbers )
• Counting  in Base Two
• Subtraction in Base Two

Number Bases (Expansion of Base Numbers )

When counting days in a week, we count in 7’s, but when counting seconds in a minute, we count in 60’s.  However, for most purposes, people count in  10’s.

The digits 0, 1,2, 3, 4, 5,6, 7, 8, 9  are used  to represent numbers.  The placing of the digits shows their value . For example,

7   8    0   9  means

• 7 thousands
• 8 hundred
• 0 tens
• 9 units

7809  = 7 x 1000 + 8  x 100  + 0  x 10  + 9 x 1

=  7  x 103  + 8 x 102  + 0  x 101 + 9 x 100

(Note : Any number raised to the power zero = 1) since the illustration above is based on the power of 10, It is called base 10. We can write it as 7809 ten

Other number systems are sometimes used. For instance 145 eight , means

• 1 eight squared
• 4 eights
• 5 units

145eight= 1 x 82  + 4 x 81 + 5 x 80

=  1  x 82  + 4 x 81 + 5 x 1

Example 1

Expand the following in the powers of their bases

1. 2389ten
2. 1001 two
3. 647eight

Solution

Using the model provided above

1. a)   2   3   8   9ten

=  2  x 103  + 3 x 102 + 8 x 101  + 9 x 100

= 2 x 103 + 3 x 102 + 8 x 101  + 9 x 1

1. b) =  1 0  0 1two

= 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20

= 1 x 23 + 0 x 22 + 0 x 21 + 1 x 1

1. c)    6   4   7 eight

= 6  x 82  + 4 x 81 + 7 x 80

= 6 x 82  + 4 x 81  + 7 x 1

Evaluation :

Expand the  following base numbers in the powers of their bases.

1. 8  1  0  6  2  nine
2. 1  0  1  1  0  1 two

1. Counting in Base Two

From the example above,  (b) was 1001 two, this means 1001 in base two.  The first thing to notice is their base two number or BINARY NUMBER, is made up of only two digits 0 and 1(just as in base ten there are ten digits: ), 1, 2, 3, 4, 5,6,7,8,9,)

In summary

Base two  ________ 0, 1

Base three ________ 0, 1,2,

Base four _________ 0, 1,2, 3. etc

The place value of the digits in the binary number  1111two is as shown below:

Eight (23)

Fours(22)

Two(21)

Units(20)

Class Activity

Work in pairs. Get a collection of about 25 counters ( e.g. matchsticks, bottle tops, smooth pebbles)

Make a paper abacus and use it to answer the following questions.

1. count out nine counters
2. group them in twos.
3. Now  group the pairs in eights, fours, twos and units as far as possible .

You will discover that nine is made up of

• 1 eight
• 0 fours
• twos, and
• 1 unit.

(d) Represent the binary number for 9 n your paper abacus.

IMPORTANCE OF BINARY SYSTEM

The binary system is second in importance to our usual base ten system. It is important because it is used in computer programs.  Binary numbers are made up of only two digits, 1 and 0. A computer contains a large number of stitches.

Each switch in either ‘on’ or ‘off’. An ‘on’ switch represents 1; and ‘off’ switch represents 0.

See the table below for the first ten binary numbers

Base ten number     Binary number

1            1

2            10

3            11

4            100

5            101

6            110

7            111

8            1000

9            1001

10            1010

Remember the following :

0  + 0  = 0

0   + 1 = 1

1  +  0  = 1

1   +  1 =  10

Example 1.

Calculate in base two

1  0 1   + 1  0   1

Solution

1  0    1

+ 1  0   1

1  0  1  0

Note: 1st  column : 1 + 1 = 0, write down  0 carry 1

2nd column : 0 + 0 + 1 carried

= 1, write down 1 carry 0

3rd column: 1 + 1 + 0 carried = 10

Example 2

Simplify the following in base two

1. a)        1   0   1   0    1

+             1   1    1

______________

1. b)        1    1     1

+              1

__________

1. c)   1    0    1

+ 1    1    0

_________

Solution

1. a)   1   0   1   0   1

+          1   1   1

11   1   0   0

Note:

1st column :1 +  1 = 10, write 0 carry 1

2nd column: 0 + 1 + 1 carried = 10,  write 0 carry 1

3rd column: 1 + 1 + 1  carried = 11, write 1 carry 1

4th column: 0 + 1 carried = 1, write 1 carry 0

5th column: 1 + 0 carried = 1

= 11100 two

Using the above explanation try out the examples worked by your teacher below:

1. b)     1    1    1

+         1

1 0  0    0

( c)            1    0    1

+    1   1     0

1  0   1     1

Evaluation:

Simplify the following in base two

1. a) 1    1    1    1

1    1    0    1

+     1    0     1

___________
b)        1   0     1

1   0     1

+  1   1     1

___________

Note: You may also need to listen to teacher’s other approach in the class to see the one you will prefer.

For instance:

4   8    9

+  3   8    2

8   7    1

This is because it is in base ten. Once, it is 10 or more than your teacher told you in addition of whole numbers that we carry. When it is less than 10 you write down the number.

The same thing is happening in base two. Once it is or  more you must carry when it is 2 you write down 0 carry 1. i.e

2

2   =  1 remainder 0

Usually, we write the remainder and carry the quotient.

See illustration

1   1   1

+     1   1

1  0  1  1

1  + 1  = 2 ( 2/2 = 1 r0 )

1 + 1 + 1 carried = 3 ( 3/2 = 1 r 1 )

1 + 1 carried = 2 ( 2/2 = 1 r 0 )

the answer =  1  0  1  0 two

Subtraction in Base two

Example 1

Simplify in base two

Solution

1   1     1

-   1   1     0

_______  1

Ans = 1 two

Example 2

Simplify in base two

1   1   1   1   0

-       1   1   0   1

1   0   0   0    1

______________

Ans = 10001 two

Example 3

Simplify in base two

1   0  0   1  1

-          1   1  0

____________

Solution

1   0   0   1   1

-          1    1   0

1  1    0   1

Ans = 1101 two

Note, the same method we used when we were subtracting whole numbers is still the method we have used.  The only difference is their bases. The whole number was in base 10.

e.g        4   8    3   - 2  9   6

4   8    3

-   2   9    6

1   8   7

In the above example, when the number we are to subtract is larger, we borrow from the next digit. For instance, we borrowed 1 from 8 reducing it to 7 and increasing 3 to 13. Each 1 borrowed is equal to 10 which represents the base.

In our own case, any 1 borrowed is equal to two representing the base.

Try your self in base eight and 1 borrowed is equal to_____

Evaluation

Simplify the following in base two

1. a) 1  0  1  1  1

-   1  0  1 1 1

________

1. b) 1  1  1  0  0

-       1  1  1  1

_____________

1. c)                   1    1    1

-       1    1

_________

i.Multiplication  in base two

1. Conversion

Weekend Assignment

1. Binary numbers means ________ numbers(a) base two (b) base ten (c ) base four (d) base eight
2. Base two numbers are made up of two digits _____ and ______

(a) 0    and 1   (b) ) and 2      (c  ) 1 and 3   (d) 0, 1 and 2

1. simplify in base two  ( 1 1 1 + 1 1 1 ) (a)1 1 0 1    (b) 1  1  1  0    ( c ) 1  0  0  1   ( d) 1  0  0 0
2. Simplify in base two  ( 110  - 11 ) (a) 11 (b) 101    (c ) 100 (d) 1
3. Expand 586 nine

(a) 5 x 92 + 9 x 81 + 6 x 90

(b) 5 x 93 + 8 x 91 + 6 x 1

(c ) 5 x 93 + 8 x 92  + 6 x 91

(d) 5 x 92 + 8 x 91 + 6 x90

Theory

Simplify the following in base two

1a    1  1  1  0

+ 1  0  0  1

__________

1. b) 1  0  1   0   1

+       1   1   1

___________

c).  1  1  0  1 two  + 1  1  0  0  1 two  + 1  0  1 1 two

2a)    1    1  0   1    1

-  1   0    1   1   1

______________

1. b) 1  0  1  1  1

-     1  0  1  1  1

_______________