Lesson Notes By Weeks and Term - Senior Secondary School 1

NUMBER BASE CONVERSIONS

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SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

  • New General Mathematics SSS 1     M.F. Macrae et al 

 

 

 
WEEK TWO

TOPIC: NUMBER BASE CONVERSIONS

People count in twos, fives, twenties etc. Also the days of the week can be counted in 24 hours. Generally people count in tens. The digits 0,1,2,3,4,5,6,7,8,9 are used to represent numbers. The place value of the digits is shown in the number. Example: 395:- 3 Hundreds, 9 Tens and 5 Units. i.e.

                 39510      = 3 x102 + 9 x 101 +5 x 100.

Since the above number is based on the powers of ten, it is called the base ten number system i.e.

                               = 300 + 90 + 5. 

 Also 4075 = 4 Thousand 0 Hundred 7 Tens 5 Units i.e. 4 x 103 + 0 x 102 + 7 x 101 + 5 x 100 Other Number systems are sometimes used. 

Example: The base 8 system is based on the power of 8. For example: Expand 6477, 265237, 1011012,

(a)    6457       = 6 x 72 + 4 x 71 + 5 X 70 = 6 x 49 + 4 x 7 + 5 x 1

(b)    265237   = 2 x 74 + 6 x73 + 5 x 72 + 2 x 71 + 3 x 70

(c)    1011012 = 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x21 + 1 x 20

 

EVALUATION

Expand The Following

  1. 7358 2.    10100112

 

CONVERSION TO DENARY SCALE (BASE TEN)

When converting from other bases to base ten the number must be raised to the base and added.

Worked Examples:

Convert the following to base 10

(a)    278    (b)    110112

Solutions:

(a)    278 = 2 x 81 + 7 x 80 = 2 x 8 + 7 x 1 = 16 + 7 = 23

(b)    110112 = 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1

                         = 16 + 8 + 0 + 2 + 1 = 27

 

EVALUATION

Convert The Following To Base Ten:

(a)    1010112        (b) 21203

 

CONVERSION FROM BASE TEN TO OTHER BASES

To change a number from base ten to another base

  1. Divide the base ten numbers by the new base number;
  2. Continue dividing until zero is reached;
  3. Write down the remainder each time; 
  4. Start at the last remainder and read upwards to get the answer.

Worked Examples:

  1. Convert 6810 to base 4.
  2. Covert 12910 to base 2

Solutions:

  1.                   5   68                                                 2   129

                        5 13 R 3                                                2   64 R 1

                        5   2 R 3                                                2   32 R 0

                    0 R 2                                                2   16 R 0

                                2335                                                  2    8  R 0       100000012

                                2    4  R 0

                                2    2  R 0

EVALUATION                            2    1  R 0

  1. Convert 56810 to base 8                                                 0  R 1
  2. Convert 10010 to base 2

 

Bicimals

Base ten fractions, or decimals, are based on negative powers of ten 

 

    6 100    5 10-1    8 10-2    3 10-3

 

                6.583

Similarly we can have base two fractions, bicimals, based on negative powers of two 

 

    1 20        1 2-1        0 2-2        1 2-3

 

                1.101

 

To convert a bicimal to a decimal, first express each digit as a power of two, then change the powers to fractions. Study the example below 

 

Example 1 

Convert the following bicimals to decimals. 

  1. 1.101 b. 10.011 c. 110.11

 

  1. 1.101    =    1 20 + 1 2-1 + 0 2 -2 + 1 2-3

=    1 + 1 12 + 0 122 + 1 123

=    1 + 12 + 0 + 18

=    1 + 0.5 + 0 + 0.125 

=    1.625

  1. 10.011     =    1 + 21 + 0 20 + 0 x 2-1 + 1 2-2 + 1 2-3

=    2 + 0 + 0 12 + 1 122 + 1 123

=    2 + 0 + 0 + 14 + 18

=    2 + 0.25 + 0.125

=    2.375

  1. 110.11    =    1 22 + 1 21 + 0 20 + 1 2-1 + 1 2-2

=    4 + 2 + 0 + 12 + 14

=    6 + 0.5 + 0.25

=    6.75

EVALUATION

Convert the following bicimals to base ten. 

  1. 10.0001
  2. 10.01
  3. 11.1 
  4. 0.001 

 

Conversion of number from one base to another base 

A number given in one base other than base ten can be converted to another base via base ten. 

 

Example 1 

Convert:    (a) 1534six to base eight 

        (b) 8A9Fsixteen to base eight. 

Solution 

  1. 1534six to base eight

First convert 1534six to base ten.

1534six    =    1 63 + 5 62 + 3 61 + 4 60

        =    216 + 180 + 18 + 4

        =    418ten

Now convert 418ten to base eight. 

 

8    418    Remainders 

8    52    2

8    6    4

    0    6                i.e. 418ten = 642eight

Thus, 1534six = 642eight

 

  1. 8A9Fsixteen to base eight 

8A9Fsixteen     =    8 163 + 10 162 + 9 161 + 15 160

        =    32768 + 2560 + 144 + 15

        =    35487ten

Now convert 35487ten

8    35487    Remainders 

9    4435    7

8    554    3

8    69    2

8    8    5

8    1    0

    0    1

 

i.e.    35487ten    =    105237eight

thus, 8A9Fsixteen    =    105237eight

 

Example 2

Determine the number bases x and y in the following simultaneous equations: 

32x – 12y = 9 ten and 23x – 21y = 4ten

 

Solution

    32x – 12y = 9ten            (1)

    23x – 21y = 4ten            (2)

Change equation (1) to base ten as follows: 

(3 x1 + 2 x0) – (1 y1 + 2 y0) = 9 

3x + 2 – y – 2 = 9

3x – y = 9                     (1a)

Similarly, change equation (2) to base ten: 

i.e.    x – y = 1                 (2a)

subtracting equations (2a) from (1a):

    2x = 8

    X = 4 

Substituting x = 4 in (2a)

    4 – y = 1 

    4 – 1 = y 

    y = 3 

Thus, x = 4 and y = 3. 

 

EVALUATION 

  1. If x represents a base number in the following equations, what is the value of x? 
    1. 315x – 223x = 72x
    2. 405x + 43eight = 184ten
  2. Convert each of the following to the base indicated: 
  1. 10401.11seven to base eight 
  2. 4B3Fsixteen to base twelve 

   

GENERAL EVALUATION

  1. Convert 

(a) 178510 to base 7        (b) Convert 21256 to base 10

  1. Determine the number bases x and y in the following simultaneous equations: 
  1. 31x + 20y = 2310

23x – 11y = 510

  1. 26x – 34y = 10002

38x – 21y = 1115

  1. Find the value of Q if (Q4)2 = 1001002

 

READING ASSIGNMENT

New Gen Math SS 1pg52 – 51 

 

WEEKEND ASSIGNMENT

  1. Express 3426 as number in base 10         (a) 134        (b) 341        (c) 143
  2. Change the number 100102 to base 10    (a) 1001      (b) 40          (c) 18
  3. Express in base 2, 10010                         (a) 100100  (b) 1100100 (c) 11001
  4. Convert 120 base 10 to base 3                (a) 111103  (b) 12103      (c) 121103
  5. Convert 25 base 10 to base 2                  (a) 110012   (b) 10012     (c) 11002

 

THEORY

  1. Convert 23647 to base 10
  2. Convert 10510 to base 2

































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