Binary Numbers: Number Base Conversions – JSS3 Mathematics Lesson Note (Third Term)

Topic: number base conversions

People count in twos, fives, twenties, etc. Also, the days of the week can be counted as 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. 3X102 + 9 X 101 +5 X 100.

Since the above number is based on the powers of tens 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. 

For Example, The base 8 system is based on the power of 8. For example: Expand 6478, 265237, 1011012,

(a)    6478  = 6 x 82 + 4 x 81 + 7 X 80 =6 x 64 + 4 x 8 + 7 x 1

(b)    265237 =2 x 74 + 6 x73 + 5 x 72 + 2x 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. 4343
2. 1011112

 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)    178    

(b)    110112

 Solutions:

(a)    178 = 1 X 81 + 7 X 80 = 1 X 8 + 7 X 1 = 8 + 7 = 15

(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)    101002

(b)    21203

 Conversion from base ten to other bases

To change a number from base ten to another base

1. Divide the base ten number 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 6
2. Covert 12910 to base 2

Solutions:

1.           6    68

                      6    11 R 2

                      6     1 R

                      0 R 1

                     = 1526

2.             2     129

                       2    64 R 1

                       2    32 R 0

                       2    16 R 0

                       2    8 R 0

                       2    4 R 0

                       2    2 R 0

                       2    1 R 0

                       2    0 R 1

                       = 100000012

EVALUATION

1. Convert 56910 to base 8
2. Convert 10010 to base 2

GENERAL EVALUATION

Convert the following to base seven

1. 405 ten
2. 876 ten

Evaluate the following

3. 538  – 314 + 412

READING ASSIGNMENT

New Gen Math Book 3  pg 15-17

Essential Mathematics for J.S.S.3 Pg 5 -9

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 12648 to base 10
2. Convert 10510 to base 2