Number Bases I

Mathematics SSS1 First Term

WEEK 1

Number Bases I

Performance Objectives

Students should be able to;

1. Mention other bases such as 4, Base 5(quandary), base 8(octal), base 16(Hexadecimal), e.t.c
2. Convert decimal fractions to base 10 and one base to another base
3. Convert numbers from base 10 to other bases and from other bases to base 10

Number Base System

Content

Concept of expanded notation: Every decimal number X can be expressed uniquely in the form:

 

X = In × 10n + In−1 × 10n−1 + In−2 × 10n−2 + ... + In−n × 10n−n

This is known as the expanded notation

Example 1: Express the following in expanded notation form

a. 45078

b. 0.0235

c. 930.133

Solution:

 

45078 = 4 × 104 + 5 × 103 + 0 × 102 + 7 × 101 + 8 × 100 = 4 × 10000 + 5 × 1000 + 0 × 100 + 7 × 10 + 8 × 1 0.0235 = 0 × 100 + 0 × 10−1 + 2 × 10−2 + 3 × 10−3 + 5 × 10−4 = 0 × 1 + 0 × 1/10 + 2 × 1/102 + 3 × 1/103 + 5 × 1/104 930.133 = 9 × 102 + 3 × 101 + 0 × 100 + 1 × 10−1 + 3 × 10−2 + 3 × 10−3 = 9 × 102 + 3 × 101 + 0 × 1 + 1 × 1/101 + 3 × 1/102 + 3 × 1/103

Example 2: Write the following in expanded notation form

(a) 32.51₆

(b) 0.1001₂

Solution:

 

32.516 = 3 × 61 + 2 × 60 + 5 × 6−1 + 1 × 6−2 = 3 × 6 + 2 × 1 + 5 × 1/6 + 1/162 0.10012 = 0 × 20 + 1 × 2−1 + 0 × 2−2 + 0 × 2−3 + 1 × 2−4 = 0 × 1 + 1 × 1/21 + 0 × 1/22 + 0 × 1/23 + 1 × 1/24 = 0 + ½ + 0/4 + 0/8 + 1/16

Definition of Number Base System

Number system is defined by the base it uses, the base is the number of different symbols required by the system to represent any of the infinite series of numbers.

A base is also a number that, when raised to a particular power (that is, when multiplied by itself a particular number of times, as in 10² = 10 × 10 = 100), has a logarithm equal to the power.

For example, the logarithm of 100 to the base 10 is 2.

Conversion from any Base to Base 10

Two digits (0, 1) suffice to represent a number in the binary system; 6 digits (0, 1, 2, 3, 4, 5) are needed to represent a number in the hexadecimal system; and 12 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(ten), B(eleven)) are needed to represent a number in the duodecimal system. The number 30155 in the hexadecimal system is the number (3 × 6⁴) + (0 × 6³) + (1 × 6²) + (5 × 6¹) + (5 × 6⁰) = 3959 in the decimal system; the number 2BA in the duodecimal system is the number (2 × 12²) + (11 × 12¹) + (10 × 12⁰) = 430 in the decimal system.

Note: To convert from any base to base ten, expand the given number(s) in the powers of their bases and simplify.

Examples:

 

1. Convert 1243five Solution: 1243five =(1×53) + (2×52) + (4×51) + (3×50) = 125 + 50 + 20 + 3 = 198ten 2. Convert 1111110two to a number in base ten. Solution: 1111110two =(1×26)+(1×25)+(1×24)+(1×23)+(1×22)+(1×21)+(1×20)=64+32+16+8+4+2+0=126ten

Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.

 Conversion from other Base Greater than ten to Base Ten

Expansion method can be used to convert numbers in base, say, base thirteen to base ten. Remember in base thirteen the digits we have are (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C). where A represents ten B represents eleven and C represents twelve. Letters are used for two- digits numbers less than the base thirteen.

Examples:

1. Convert 1B9_thirteen to denary number

Solution

1B9_thirteen= 1 × 13² + B × 13¹ + 9 × 13⁰

= 1 × 169+ 11 × 13 + 9 × 1

= 169 + 143 + 9

= 321_ten

2.  Convert 20C_fifteen to a denary number

Solution

20C_fifteen= 2 × 15² + 0 × 15¹ + 12 × 15⁰

= 2 × 225 + 0 × 15 + 12 × 1

= 450 + 0 × 15 + 12 × 1

= 462_ten

 

Conversion of Numbers From one Base to Another Base

To convert from a base to another you may have to pass through base ten.

Examples

1. Convert 301_four to a base six number.

Solution

First 301_four will be converted to a base ten number

301_four = 3 × 4² + 0 × 4¹ + 1 × 4⁰

= 48 + 0 + 1

= 49_ten

 

 

 

49_ten will now be converted to a base six number by repeated division

 

6	49
6	8	R	1
6	1	R	2
	0	R	1

                                                                                                                          301_four = 121_six

2. convert 110111₂ to base 5.

Solution

110111₂ = 1 × 2⁵+ 1 × 2⁴ +0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

= 1 × 32 + 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 1 × 1

= 32 + 16 + 0 + 4 +2 +1

= 55₁₀

Then we convert 55₁₀ to a number in base 5

 

5	55
5	11	R	0
5	2	R	1
	0	R	2	↑

                                                                                                                         Hence, 55₁₀ = 210₅

 

Simple Equations in Number System

Simple linear equations and simultaneous equation can be solved using the knowledge of expansion to base 10.

 

Examples If x10 = 12145, find x. Solution X × 100= 1 × 53 + 2 × 52 + 1 × 51 + 4 × 50 X = 125 + 50 + 5 + 4 X = 184   If 55x + 52x = 7710, find x. 5 × x1 + 5 × x0 + 5 × x1 + 2 × x0 = 77 5x + 5 + 5x + 2 = 77 10x + 7 = 77 10x = 70 X = 7