Week 4 – Jss 1 Second Term Mathematics Lesson Notes

 WEEK 4:            NUMBER BASE
4.0    Introduction:
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. 0,1,2,3,4,5,6,7,8,9.

 4.1    Expansion and Conversion to Base Ten
Expanded Notation
Example 4.1: Write the following in expanded nation form.
(a) 1101100₂
(b) 2135₆
(c) 4567₈

 Solution:

1. 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).

 
 4.2    Conversion to Base Ten
Example 4.2: convert the numbers in example 1 to Base ten
Solution:
In other to do this we simply continue from the expanded notation, evaluate and get our answers.

1. 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

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

= (2x 216) + (1x 36) + (3x 6) + (5 x 1)
= 432 + 36 + 18 + 5 = 491

1. 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

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

 
 Converting Number in Base Ten To Numbers In Base 2
Examples4.3:(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 1            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

 ASSIGNMENT: PAGE 105
EXERCISE 9.2 NO 3 (e,f,g,h) , NO 5 (a,b,d)