Number Base JSS1 Mathematics Lesson Note

The usual system of counting in our days is called the decimal or denary system. The denary or decimal system is also called base ten. This system enables us to be able to write small or large numbers using the combination of the digits, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Expansion and Conversion to Base Ten

Expanded Notation

Example 4.1: Write the following in expanded notation form.

(a) 11011002

(b) 21356

(c) 45678

Solution:

(a) 16150413120100= (1 x 26) + (1 x 25) + (0 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (0 x 20).

 The base is used to expand it and the power for each expansion.

(b) 23123150 = (2 X 63) + (1 x 62) + (3 x 61) + (5 x 60)

(c) 43526170 = (4 x 83) + (5 x 82) + (6 x 81) + (7 x 80).

Conversion to Base Ten
 Convert the numbers in example 1 to Base ten

Solution:

In order to do this we simply continue from the expanded notation, evaluate, and get our answers.

(a) 16150413120100=(1×26)+(1×25)+(0x24)+(1×23)+(1×22)+(0x21)+(0x20).

                              = (1 x 64) + (1 x 32) + (0 x 16) + (1 x 8) + (1x 4) + (0 x 2) + (0 x 1)

                              = 64+32+0+8+4+0+0 = 108

(b) 23123150=(2X63)+(1×62)+(3×61)+(5×60)

                     = (2x 216) + (1x 36) + (3x 6) + (5 x 1)

                     = 432 + 36 + 18 + 5 = 491

(c) 43526170=(4×83)+(5×82)+(6×81)+(7×80).

                    = (4 x 512) + (5 x 64) + (6 x 8) + (7×1)

                    = 2048+320+48+7=2423

 Binary System 

In a binary system, the greatest digit used is 1, so the two digits available in a binary system are 0 and 1. Remember that each digit in a binary number has a place value.

Converting Numbers in Base Ten To Numbers In Base 2

Examples :
(a) Convert 2910 to base 2.

(b) Convert 7910 to base 2

(c) convert 14510 to base 2

Solution: 

(a). 2    29 (b). 2 79 (c) 2 145

2 14 R 1    2 39 R  2 72 R 1

2 7 R 0 2 19 R 1 2 36 R 0

2 3 R 1 2 9 R 0 2 18 R 0

2 1 R 0 2   4 R 1 2   9 R 0

0 R 1 2   2 R 0 2   4 R 1

2   1 R 0 2   2 R 0

  0 R 1 2   1 R 0

  0 R 1

2910 = 101012, 7910 = 10010112, 15710 = 100100012

EVALUATION: 

(1) Expand the following with their bases.

(a)     35318

(b) 1010102

(c) 1110110242

EVALUATION: 

(2) Convert the following number to base 2

1. 35610
2. 4710
3. 21810