Base numbers: Base Two

Content

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

  145_eight= 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

a)   2   3   8   9_ten

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

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

b) =  1 0  0 1_two

= 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰

= 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 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

II. 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

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

3. 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

4.  Simplify in base two  ( 110  – 11 ) (a) 11  (b) 101 (c ) 100 (d) 1

5. 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