CHAPTER ONE

Chapter 1: NUMBER BASES

Learning objectives

By the end of this topic, the learners should be able to

• Identify numbers in any base using abacus

• Convert numbers from one base to another

• Manipulate numbers in different bases with respect to all four operations

• Identify place value in different bases

Introduction

[1]A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0. The most commonly used number system is the decimal system, commonly known as base 10. In everyday life, we count or estimate quantities using groups of ten items or units. This may be so because, naturally, we have ten fingers. For example, when we count ten, i.e. we write 10 meaning one group of 10 and no units. A quantity like twenty five, written as 25 means 2 groups of 10 and 5 units Suppose instead we had say 6 fingers

• How, in your opinion would we do our counting?

• If we had eight fingers, how would we count?

This is now what we are to cover under this topic.

NOTE

• The digits of a number in any base are less than the base itself

• The digits 10 and 11 are represented by t and e respectively in number bases

• For digits above 11 are represented by alphabetic letters of your choice

• The names of some number systems is as given below

Activity:Getting familiar with number bases

Bases are used in day today life. Therefore copy and complete the table below by giving some real life situations were bases are used

1.1 Identifying numbers of different bases on an abacus

1. Which possible base does each abacus below represent.

(b) Write down the numbers represented on the abaci above.

2. Write down the numbers represented on the abaci below.

Activity: List the numerals for the following bases

Numerals are digits(or symbols) that are used for writing numbers in a given base. The digits are always less than the base itself. study the table below and fill in the gaps.

1.2 Place Values Using the Abacus

The representation of numbers on an abacus helps in identifying the place value of digits in any base.

Activity : Making abaci

1. Make abaci for the following number bases.

(a) 220four

(b) 6321ten

(c) 43five

(d) 1110two

(e) 5552six

(f) 6431seven

(g) 7562eight

(h) 654nine

(i) 5974eleven

2. Make an abacus for any number in base twelve

Activity : Reading and stating the value of digits in bases

1. State the place value and value of digit for each numeral in the following numbers:

(a) 523

(b) 5:12ten

1.1 Exercise Set

1. State the place value of each numeral in the following numbers:

(a) 143five

(b) 110two

(c) 5552six

(d) 431seven

(e) 4562eight

(f) 4:1234ten

(g) 45:62eight

(h) 456:212ten

2. State the value of each numeral in the following numbers:

(a) 1432five

(b) 111two

(c) 3412six

(d) 431seven

(e) 6542eight

(f) 4.1234ten

(g) 45.62eight

(h) 456.212ten

1.3 Converting Numbers

Numbers can be converted from one base to another, and when you do this, you get the same numbers written in different bases.

1.3.1 Converting from any base to base ten

EXAMPLES

Convert the following to base ten

1. 222four

2. ee0twelve

3. 1075eight

4. 45.4eight

1.2 Exercise Set

1. Convert the following numbers to base ten

(a) 1432five

(b) 111two

(c) 3412six

(d) 431seven

(e) 6542eight

(f) 1202three

(g) 321four

(h) 4518nine

2. Convert 68.3nine to base ten

3. Convert the following binary numbers to base 10:

(a) 110

(b) 1111

(c) 1001

(d) 1101

(e) 10001

(f) 11011

(g) 1111111

(h) 11001101

(i) 111000111

4. A particular binary number has 3 digits.

(a) What are the largest and smallest possible binary numbers?

(b) Convert these numbers to base 10.

1.3.2 Converting from base ten to other bases

• We use BNR

• Divide the number repeatedly by the required bases

• The remainder in reverse order gives the required number

1. Convert 19ten to base two

2. Convert 85ten to base eight

3. Convert 762eight to base seven

4. Convert 32ve to base two

5. Convert 5432six to base twelve

1.3 Exercise Set

1. Convert the following numbers to the bases indicated:

(a) 19 base two

(b) 568 to base nine

(c) 1256 to base eleven

(d) 6410 to base three

(e) 27ten to base eight

(f) 246ten to base ve

(g) 20twelve to binary

2. Convert the following numbers to the bases indicated:

(a) 34ve base two

(b) 568nine to base eleven

(c) 111two to base four

(d) 234ve to base nine

(e) 647 to base three

(f) 276eight to base twelve

(g) 341ve to base six

(h) tt5eleven to base twelve

(i) 5432six to base twelve

(j) 554six to base four

1.4 Operation on Numbers in Various Bases

In this section we are going to look at the four mathematical operations which include addition, subtraction, division and multiplication

Activity: James had two jackfruit trees in his compound. At one time one tree had 8 fruits ready and the other 7 fruits. He harvested them at the same time.

• If james puts the jack fruits in heaps of ten fruits. How many heaps of ten did he get and how many remained?

• If james puts the jack fruits in heaps of nine fruits. How many heaps of nine did he get and how many remained?

• If james puts the jack fruits in heaps of ve fruits. How many heaps of ve did he get and how many remained?

When you put the fruits in heaps of 10,9 and 5, you are adding in base 10,base 9 and base 5.

1.4.1 Addition of bases

• If the sum of the digits exceeds the base, divide that sum by the base then write down the remainder and carry the whole number.

EXAMPLES

1. Workout 234 five + 23 five leaving your answer in the base indicated

1.4 Exercise Set

1. Workout the following leaving your answer in the base indicated

(a) 232 six + 451 six

(b) 66 seven + 56 seven

(c) 11101 two + 11010 two

(d) 577 eight + 165 eight

(e) 999 ten + 245 ten

(f) 684 twelve + 436 twelve

(g) 36:64 nine + 4:31 nine

(h) 66.45 eleven + 4.65 eleven

2. Workout 233four + 544six giving your answer in base ve

3. Calculate the binary numbers:

(a) 111 + 101 + 100

(b) 11101 + 10011 + 110111

4. Workout the following leaving your answer in the base indicated

(a) et4twelve + tt3twelve

(b) 234ve + 413ve

1.4.2 Subtraction of bases

• In case of borrowing the new value is the sum of the base and the digit which was small

1.5 Exercise Set

1. Workout the following leaving your answer in the base indicated

(a) 1022 three - 210 three

(b) 31 eight - 17 eight

(c) 11111 two - 1010 two

(d) 577 eight - 165 eight

(e) 999 ten - 245 ten

(f) 684 twelve - 436 twelve

(g) 36:64 nine - 4:31 nine

(h) 66:45 eleven - 4:65 eleven

2. Subtract the following numbers in the given bases:

(a) 354 six from 553 six

(b) 845 twelve from t43 twelve

3. Workout 221three - 111two giving your answer in base ve

4. Workout 567eight - 146seven giving your answer in base six

5. Find the value of n ,45n = 29

1.4.3 Multiplication of bases

• Find the product of any two numbers as we do in base ten

• Divide this product by the base number

• Write the remainder and carry the quotient to the next place value position

EXAMPLES

1. Workout 136seven × 4seven leaving your answer in the base indicated

1.6 Exercise Set

1. Fill in the missing numbers in this multiplication table in base twelve

2. Workout the following leaving your answer in the base indicated

(a) 1022 three × 21 three

(b) 315 eight × 17 eight

(c) 1111 two × 10 two

(d) 577 eight × 165 eight

(e) 999 ten × 245 ten

(f) e84 twelve × et twelve

(g) 3664 nine × 31 nine

(h) 8745 eleven × t65 eleven

3. Multiply:

(a) 1121 three by 212 three

(b) 312 four by 122 four

(c) 41 five by 12five

(d) 1001 two by 11 two

4. Multiply:

(a) 11 two by 12 three

(b) 8787 nine by 435 seven

(c) 231 four by 235 six

(d) 1123 four by 234 five

(e) tet4 twelve by t67 eleven

(f) 3421 five by 133 four

1.4.4 Division of bases

• Convert each number base to base ten

• Divide the two numbers in base ten

• Convert the result back to the required base

EXAMPLES

1. Divide 1331four by 121four

First Convert 1331four and 121four to base ten and then nally express the answer in base

2. Divide t46eleven by 26eleven

First convert t46eleven and 26eleven to base ten and then finally express the answer in base eleven .

Converting t46eleven to base ten