Topic 1: Number Bases

By the end of this topic, you will be able to:

• identify numbers in any base using an abacus
• convert numbers from one base to another
• manipulate numbers in different bases with respect to all the four operations
• identify place values in different bases

Keywords

• abacus
• base
• number
• operation
• place value

Introduction

Look around you, what kind of number can you read on the charts, calendars, clock, computers and any other material? All the numbers are either 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or a combination of these symbols.

Do you notice that the biggest digit is 9? Other numbers such as 10, 11, 13…. are a combination of the digits 0 to 9.

Think about it!

What if there were other symbols for 10, 11, 12, … how many symbols would you need to count to a million?

How is 50 and 20 added in these two cases? Why does the first case give 10 and the second case gives 70?

But not: 12:50 + 20 12:70 ?

Supposing the clock never stopped at 12 hours or 24 hours, how big would a clock face be to measure 10 days? Or if a calendar never stopped at 12 months, how big would the calendar be to measure your age? Number bases are used to count items in simpler divisions. In this topic, you will learn how to use decimal place values to develop understanding of numbers written in other bases.

Identifying Numbers in any Base using an Abacus

Activity 1.1 Identifying number bases in real life (work in groups)

What you need: a pen, notebook What to do: Copy and complete the table below. Table 1.1

Identify any other real life situations where bases are used. Present your work to the class.

Activity 1.2 Stating digits used in a given base (work in groups)

What you need: notebook, pen What to do:

1. Copy and complete the table below;

Table 1.2

2. Record a conclusion you can make from table 1.2 above?

3. Suggest a kind of digits you can use to represent base ten. Suppose you were given t to represent 10, and e to represent 11. List the digits for bases 11, 12.

4. Present your work to the class.

For a given number in any base, you can use the abacus to break it down, depending on the base. For example, in the number 647, how many ones, tens and hundreds are there?

Activity 1.3 Identifying numbers in any base using an abacus (work in groups)

What you need: counters, pen, an abacus

What to do:

1. Pick a number of counters and group them in numbers of your choice such as tens, fives, fours, etc.

2. Represent the groups on an abacus.

3. Write the number you have shown on the abacus for each.

4. Present your work to the whole class.

Example (Discuss in groups)

Write down the number which has been represented on each of the abacuses below.

Exercise 1.1

1. Identify the numbers shown on each of the abacuses below.

2. Show each of the following numbers on the abacus.

a) 2345 eight b) 434 seven c) 51314six d) 123, four

3. John is supposed to give change of UGX 57,800 to his client. Using an abacus, help John to identify the number of notes of UGX 10,000, UGX 1,000 and coins of UGX 100 he will give to his client.

Identifying Place Values in Different Bases

In the figure, which number do you obtain when you interchange 2 and 3? Which number do you obtain when you move 9 to the left end? In the following activity, you will form numbers from digits and also observe how numbers change when digits change places.

Activity 1.4 Identifying place values in different bases (work in groups)

What you need: manilla papers, pair of scissors, markers, notebook and pen. What to do:

1. Prepare 4 strips numbered 0 to 9 as shown.

2. Thread them through a card such that a single number on each card is displayed.

3. Make different numbers by changing the length of the strips.

4. Write down the number which has been displayed.

5. In which base is the number?

6. Write down the place value of each digit in the number displayed.

7. Remove the last digit from each of the strips.

8. Repeat steps 3 to 6.

Every digit in a number has a place value. A Place value is the value represented by a digit in a number on the basis of its position in the number. The value of a digit is the product of that digit and its place value.

Example 1 and Example 2

What is the place value of each digit in:

Exercise 1.2

1. Identify the place values of each digit in each of the following numbers.

a) 2122 eight b) 1011, two

2. Find the value of each digit in:

a) 2356 eight b) 1034 five c) 7146 nine c) 7143. nine d) 411012 five

Converting Numbers from One Base to Another

Activity 1.5 Converting from one base to another (work in groups)

What you need: notebook and pen

What to do:

1. Find your age in years, months, weeks and days.

2. Discuss your results with other colleagues in the class.

3. Present your work to the class. Converting Numbers from Other Bases to Decimal Base

Activity 1.6 Converting from base five to base ten (work in groups)

What you need: counters such as bottle tops, notebook and pen.

What to do:

1. Make 2 groups of five and 1 group of 4 bottle tops.

2. State the number of groups of fives, and what the remainder is.

3. State the equivalence of the bottle tops in base five.

4. State the number of bottle tops they are altogether.

5. Present your work to the whole class. You can also use calculation method to convert numbers from other bases to base ten

Example 1

Number game

Supposing you are given four boxes containing numbers in base ten. The boxes are labelled box 1, box 2, box 3 and box 4. For example, you have numbers as shown in the following boxes.

Box 1 8 15 14 9

Box 2 15 14 17

Box 3 15 14 7

Box 4 15 9 7 1

1. Use digit 1 to show that the number was in the box and 0 to show that it was not.

2. Write out the numbers using the digits above.

3. In which base have the digits been written? How do you tell?

Exercise 1.3

1. Convert the following to decimal base.

a) 1111 two

b) 123 four

c) 444 five

d) 234 seven

e) 3047 nine

f) 7803 nine

g) 4512 six

h) 8301 nine 111 two three’

Converting from a Decimal Base to Other Bases

Activity 1.7 Converting from base ten to other bases (work in groups)

What you need: counters, pen, papers

What to do:

1. Get 21 counters, and group them into fives.

2. Write down the groups of fives, and ones you get.

3. Write 21… in base five. ten

4. Repeat the steps 2 to 4 by grouping into ‘fours, ‘threes’ and ‘twos.

5. Present your work to the class.

Example 1

Example 2

Exercise 1.4

1. Convert the numbers 5684, 2393, 142, 132 and 8309 to the bases indicated. a) Base two b) Base seven c) Base nine d) Base four

2. Kate buys 30 heaps of tomatoes, each with 8 tomatoes from a farmer.

a) What is the number of tomatoes in base eight?

b) By converting to base ten, how much will Kate earn by selling each tomato at UGX 200?

3. If you counted using only nine digits {0, 1, 2, ….) rather than ten.

a) What base would that be?

b) List the first 20 numbers of the arithmetic.

c) Discuss how you arrived at your answers.

Converting from a Non-decimal Base to Another Non-decimal Base

Akello buys 4 heaps, each having 9 mangoes. She wants to sell them in heaps of threes. How many heaps will she get? You can use the knowledge of base conversions to solve such problems.

Activity 1.8 Converting from a non-decimal base to another non-decimal base (work in groups)

2. Put the counters together and count them normally. How many are they? Therefore 33 four = ten

3. Group the same counters into ‘fives: How many groups do you get?

Therefore, you can convert numbers from any base to another base. Repeat the same procedures above by grouping the same number of counters into other groups.

Example

Exercise 1.5

1. Convert 115,ix to the following bases a) two b) five c) eight

Manipulating Numbers in Different Bases with Respect to all the Four Operations

Assume Peter has eight mangoes and Jamirah has six mangoes. They agree to sell them in heaps of four. How many complete heaps will they get from their mangoes? In this section, you will manipulate numbers in different bases with respect to all the four operations.

Adding and Subtracting Number Bases

Activity 1.9 Adding and subtracting number bases (work in pairs)

What you need: counters, papers, rulers, pens.

What to do: After a math lesson, Juma and Esther chose to design an addition table in base 5, as shown below.

4. Help Juma and Esther complete the table.

5. Repeat step 1 using subtraction and fill the table.

6. What steps have you gone through to complete the tables?

7. Compare your work with other pairs.

Exercise 1.6

a) Add 5472, nine to 2736, nine c) Subtract 1213 four, from 2131four,

b) Add 212 three, to 1021 three d) Subtract 512 six from 4125 six,

Multiplying and Dividing Number Bases

Multiplying number bases What is: 2 × 3 = ? 4 × 15 = ? You can multiply numbers in other bases in the same way as in base ten.

Activity 1.11 Dividing number bases (work in groups)

What you need: Pen, notebook,

What to do:

1. Convert 110111two and 101two to the decimal base.

2. Divide the results in the order given. X 32 Five

3. Convert the final result back to base two. What result do you obtain?

4. Compare your work with other groups.

Exercise 1.8

Divide the following numbers respectively and give your answer in the base given in brackets.

1. 202 three by 101,

2. 424 by 14 eight three (binary) (quinary) eight

3. Divide 44v+ 11two (octonary) five

4. Divide 200, ÷ 11 two (senary) three