Mathematics

Key unit competence

By the end of this unit, the learner should be able to present number bases
and solve related problems.

Unit Outline
• Definition of number bases.
• Change of base.
• Operations using bases (addition,
subtraction, division and
multiplication)
• Special bases (binary and
duodecimal systems)
• Solving equations involving different bases.

Introduction

Unit Focus Activity

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

(i) How, in your opinion would we do our counting?

(ii) If we had eight fingers, how would we count?

(iii) Demonstrate symbolically how counting in groups of 3, 4, 5, 6, 7…
can be done.

(iv) Do you think we could also do operations such as addition,
subtraction, multiplication and division using such groups? If your
answer is yes, demonstrate this with simple examples.
In this unit, we will learn a number of different numeration systems including
the decimal (base ten) system that we are all familiar with. We will also learn how
to convert between different numeration (counting) systems.

2.1 Numbers and numerals

Activity 2.1

Use a dictionary or internet to define:
(i) Number

(ii) Numeral

(iii) Digit

In mathematical numeral systems, we use basic terms such as number,
numeral and digit. In order to deal with number bases, we must be able
to distinguish between the three terms.

A number is an idea, a numeral is the
symbol that represents the number. The number system that we use today is a
place value system. A unique feature of this system is that the value of any of the
digits in a number depends on its position. For example the number 7 707 contains
three sevens, and each of them has a particular value as shown in table 2.1.

The 1st seven from the right represents
7 ones or units. The 2nd seven stands for 100s or 102 and the 3rd seven stands for
1 000s or 103.

The zero holds the place for the tens (10s) without which, the number would be 777
which is completely different from 7 707.

A digit is any numeral from 0 to 9. Anumeral is made of one or more digits.
For example, number one hundred and thirty five is represented by the numeral
135 which has three digits 1, 3 and 5. The number 7 707 contains four digits,
each of which has a specific value depending on its place value.

The abacus

Activity 2.2

1. Use a Mathematics dictionary or internet, to describe an abacus.

2. Describe how the abacus is used to count in base ten.

One device that has been used over time to
study the counting in different numeration
systems is the abacus.
An abacus is a calculating device consisting of beads or balls strung on wires or rods
set in a frame. Fig. 2.1, shows a typical abacus on which the place value concept
can be developed very effectively.

On each wire, there are ten beads. Let us
consider the beads at the bottom of the
wire. Beginning from the right:

10 beads on wire 1 can be represented by
1 bead on wire 2. Similarly, 10 beads on
wire 2 can be represented by 1 bead on
wire 3 and so on.

This means:
1 bead in wire 1 represents a single bead.
1 bead in wire 2 represents 10 beads.
1 bead in wire 3 represents
(10 × 10) beads.
1 bead in wire 4 represents
(10 × 10 × 10) beads.
So, the number shown in Fig. 2.1 is 1 124.
If we had x beads in each wire such that
x < 10, it would mean that:

In wire 1 we had x beads
In wire 2 we had 10x beads
In wire 3 we had 10²x beads
In wire 4 we had 10³x beads and so on.
The place values from right to left are
10⁰   10¹    10²    10³       10⁴ ...
Ones 10s 100s 1 000s 10 000s etc

2.2 Number bases

2.2.1 Definition of number bases

Activity 2.3

1. Use a dictionary or internet to find
the meaning of number bases.

2. Give some examples of numberbases.

Why do you think we count in groups of
ten?If we had 6 fingers, most probably we
would count using groups of 6, if 8 fingers,groups of 8 and so on. In the system that
we use, every ten items make one basic group which is represented in the next
place value column to the left as shown
 in Fig. 2.2 below.

(a) The 1 bead in wire B represents 10 beads in wire A i.e. it represents a
group of 10 beads.

(b) The 1 bead in wire D represents 6
beads in wire C, thus making a groupof 6 beads.

Counting in different groups of numbers
such as 10, 6, 5, 8 etc means using
different number systems. We call them
base ten, base six, base five, base eight
respectively etc.
Now consider Fig. 2.3.

Counting in base six, what numbers do
the beads on each wire represent?
(i) There are 4 beads in wire A. This
    represents 4 ones.

(ii) There are 5 beads in wire B. This
    means 5 groups of 6 beads each.
    i.e. 5 × 6 = 30 beads written as 50six
.
(iii) There are 3 beads in wire C. This
     means 3 groups of six sixes i.e.
     3 × 6 × 6 = 108 beads, written as
     300_six
.
(iv) There are 2 beads in wire D. This
means 2 groups of six six sixes ie
6 × 6 × 6 = 216 × 2 = 432
written as 2 000_six
.
The whole number represented in Fig. 2.3
is 4_six + 50_six + 300 _six + 2 000 _six = 2 354_six
The answer 2 354 _six is read as; two three
five four base six. The number 2 354six has
a value of 575_ten.

Example 2.1

Given that the number represented in
Fig. 2.4 is in base six, find the number in
base 10.

Solution
Column A represents 3 ones.
Column B represents 5 sixes.
Column C represents 2 six sixes.
the number = ( 3 × 60) + (5 × 6) +(2 × 62)
                    = 3 + 30 + 72
                    = 105ten
                    253_six = 105_ten
Note that 253_six and 105_ten are two different
symbols for the same number.

2.2.2 Change of base

(a) Changing from base 10 to any
other base

Activity 2.4

Consider the number 725 given in
base ten.
1. Divide 725 by 8 and write down
     the remainder.

2. Divide the quotient obtained in (1) above and write down the
    remainder.

3. Repeat this process of division by 8 until the quotient is less than
    8 which you should treat as a remainder and write it down.

4. Write down the number made by the successive remainders
    beginning with the first one on the right going left.

5. Describe the number in part (4) above in terms of a base

 In this activity, you have just converted 72510 to a number in base 8.

In converting
any number from base ten to any other base, we use successive division of the
number by the required base. The new number is obtained by writing down
the remainders beginning with the
first remainder on the right to the last
remainder on the left.
For example, to change 42510 to base 6,
we do successive division by 6.
 425 ÷ 6 = 70 Rem 5
 70 ÷ 6 = 11 Rem 4
 11 ÷ 6 = 1 Rem 5
 1 ÷ 6 = 0 Rem 1
The successive remainders read upwards
form the number 1545.

∴ 425₁₀ = 1 545₆

Exercise 2.1
1. Convert the following numbers from
    base 10 to base 5.
(a) 50    (b) 36       (c) 231

2. Convert the following numbers in
   base 10 to base 9.
(a) 82                    (b) 190

(c) 144                   (d) 329

3. Convert the following numbers in
base 10 to specified base.
(a) 145 to base 2

(b) 5204 to base 6

(c) 800 to base 2

(d) 954 to base 8

(e) 512 to base 3

(f) 1280 to base 12

(g) 896 to base 16

(b) Converting any base to base 10

Activity 2.5
Consider the number 125 given in
base six.
Using number place value method;

(a) Find the value of digit 1, 2and 5.

(b) Add up the values obtained in part (a) above.

(c) What does this value represent?

In this activity, you have converted a
number from base 6 to base 10. To
convert a number from one base to base
ten, we use number place values. For
example to convert 2539 to base 10, we
say:
2359 means 5 ones + 3 nines + 2 nine
nines.

∴ 2359 = (5×90) + (3×91 + (2×92)
            = (5 × 1) + (3 × 9) + (2 × 92)
            = 5 + 27 + 162
             = 194
∴ 2359 = 19410

(a) Consider Fig 2.5 below.

Suppose in Fig 2.5, each spike is designed to hold six beads, and that
each bead in spike B represents six beads in spike A. Thus in Fig 2.5 (b)
there are two beads in B and three beads in spike A. The 2 beads mean
2 groups of six i.e 2 × 6 or 12 beads. The 3 beads are said to represent 3
ones. Thus the number represented in Fig 2.5(b) is written as 23six
Therefore, 23six = 15ten This is read as two three base six
equal one five base ten: 23six and 15ten are different numerals
for the same number
(b) Now consider Fig 2.6 below:

The number shown in Fig 2.6 can be written as 134_six. What does the single

bead in spike C represent? It is the same as six beads in spike B which is
equal to six × six (or thirty six) beads
in spike A.
Hence 134six means:
The 1 stands for 1 six sixes or 36_ten
The 3 stands for 3 six or 3 × 6_ten
The 4 stands for 4 ones or 4_ten
So we would write
134_six as (36 + 18 + 4_ten) = 58_ten
i.e 134_six = 58_ten
(c) Now consider the number represented
in Fig 2.7 below.

When reading off a number in base six, it may help us to think in powers
of six. The number represented in
Fig 2.7 can be written as
145_six = 1×6 sixes + 4 sixes + 5ones
          = (1 × 6²) + (4 × 6) + (5 × 1)
          = 36 + 24 + 5
          = 65_tens

(d) Now, let us think of a number like
    28_ten. How can we represent 28 on a base six abacus?
   We find the number of sixes contained in 28.
   To do this we divide 28 by 6. Thus
   28 ÷ 6 = 4 Rem 4.
  So, 28ten is 4 sixes and 4 ones.
  This number can be written on the

 abacus as shown in Fig. 2.8.

i.e on the abacus, there are 4 beads
 on spike A and 4 in spike B. i.e
 28_ten = 44_six
 Use a similar method to show on a base
 six abacus the following numbers.
 81ten and 324_ten
 Note:
 We can use a similar method to represent
 any base ten number in another base.
 Also, a number such as 65_ten can be
 expressed as a number in base six as:
 65 ÷ 6 = 10 Rem 5 5 ones
10 ÷ 6 = 1 Rem 4 4 sixes
1 ÷ 6 = 0 Rem 1 1 × 6 sixes

The answer is then written starting with
the last remainder, followed by the next
remainder, etc vertically up till the first
remainder.
65_ten = 145_six

Example 2.4
Express 415six as a number in base ten.

Solution

We use place values to change from base
six to base 10.

415six = (5 × 1) + (1 × 6) + (4 × 62)
             = 5 + 6 + (4 × 36)
             = 5 + 6 + 144
             = 155
∴ 415six = 155ten

Exercise 2.2

1. Convert the following numbers from
   specified base to base 10.
  (a) 85₉                 (b) 1001₂

(c) 2343₅               (d) 12₃

(e) 615₇                (f) 142₅

(g) 1232₄

2. Are the following valid or invalid?

(a) 123₂         (b) 234₅

(c) 1002₂        (d) 3467₆

(c) Converting from one base to any other base

Suppose we wish to change from base m
to base n where m ≠ n ≠ 10 and m and n
are positive numbers.

Activity 2.6

Consider the number 46₇.

(a) Convert 46₇ to a number in base 10 as you did in activity 2.5

(b) Use your answer to part (a) above and convert it to a number in
base 5.

(c) Describe the procedure of converting a number from a
number in base x to a number in base y where x ≠ y.

In this activity, you just converted a
number from base seven to base five.

To convert a number from a base other than ten to another base, follow the steps
below.
(i) Change or convert the given number to base 10.

(ii) Convert the result of part (i) to a number in the required base, for

example,To convert 1213 to base 4;
Convert 121₃ to base 10.
Thus 121₃ = 1 + 2 × 3 + 1 × 3²
                  = 1610
Then convert 16₁₀ to base 4, by
successive division by 4.
16 ÷ 4 = 4 Rem 0
4 ÷ 4 = 1 Rem 0
1 ÷ 4 = 0 Rem 1
121₃ = 100₄
Now let us repeat activity 2.6 using
386_nine
• 386_nine means 6 ones, 8 nines and
3 nines
We first change 386nine to base ten as
follows:
386₉ = 6 × 1 + 8 × 9 + 3 × 92
        = 6 + 72 + 243
        = 321_ten
• To convert to base 6, we do successive
division of the number in base 10.

Note: to convert a number from a base
other than 10 to another base, we first
convert from the given base to base 10.
Then from base 10 to the required base.
Example 2.5
Convert 5148 to base 9.
Solution
To convert from base 8 to base a,

(i) First convert to base 10

(ii) Then convert result (i) to base 9
5148 = 4 × 1+ 1 × 8 + 5 × 82
         = 4 + 8 + 320
         = 33210
To convert 33210 to base 9, we do
successive division by 9, noting the
reminder at each step.

From down upwards the reminders form
number 408.
This means 8 ones
                   0 nines
                  4 nine-nines
Thus 514₈ = 408₉.

Exercise 2.3

1. Convert the following to base 7.

(a) 411₅      (b) 321₆

(c) 15₆     (d) 302₄

2. Express 63₇ to base 5

3. Given that 85₁₀ = 221_x. Find the
     value of x.

4. Convert the number 703₈ to;

(a) Base 6           (b) Base 10

(c) Base 9           (d) Base 2

In short;
To convert from base ten to
another base:
1. Do successive division by the required base noting the
remainders at every step.

2. Write down the remainders beginning with the last one on the
        left.

3. These remainders make up the required number.
To convert from any base x to base 10:

1. Multiply every digit in the number by its place value i.e. 1, x, x², x³ etc.

2. Add the results. To convert from base m to base n,
where m ≠ 10 and n ≠ 10:

1. First convert from base m to base 10:

2. Then, convert from base 10 to base n.

Numbers in other bases can be expressed
in the same way as we have done.
The following are some other bases
and the numerals used.

Base Numerals
Nine 0 1 2 3 4 5 6 7 8
Eight 0 1 2 3 4 5 6 7
Seven 0 1 2 3 4 5 6
Six 0 1 2 3 4 5
Five 0 1 2 3 4
Four 0 1 2 3
Three 0 1 2
Two 0 1
and so on.

(e) Base 4

In any base, the numeral equal to the base
is represented by 10.
i.e. 5₅ = 10₅     6₆= 10₆   10₁₀= 10
8₈ = 10₈ etc
When a base is greater than 10, say 12,
we need to create and define a symbol to
represent 10 and 11.

Exercise 2.4

1. Write the first twenty numerals of:
(a) Base six        (b) Base seven

(c) Base eight

2. What does 8 mean in:

(a) 108_ten        (b) 180_ten

(c) 801_ten         (d) 88 801_ten

3. Write down in words:
(a) 203_six          (b) 302_four

(c) 15_six            (d) 3 21_5eight

4. Convert the number 703_eight to:

(a) base 6         (b) base 10

(c) base 9

5. Convert the following into decimal
system:

(a) 411_five           (b) 321_six

(c) 207_eight        (d) 750_nine

6. Express 63_seven to base 5

7. Write in words the meaning of :

(a) 12_three           (b) 21_four,

(c) 142_five,           (d) 180_nine

8. Use abacus to show place values for the numerals in:

(a) 211_five               (b) 615_seven

(c) 173_eight             (d) 1 254_ten

9. Convert 118_nine to base 5.

2.3 Operations using bases

2.3.1 Addition and subtraction

Activity 2.7

Table 2.2 shows part of the addition
table for numerals in a certain base

(i) State the base.

(ii) Copy and complete the table.

(iii) Use your table to evaluate_.

Now consider table 2.3.

• Identify the base used in this table.

• Copy and complete the addition.

Table 2.3.

• List the numerals used in this table.

• Use your table to formulate some
equations involving subtraction.

Note:
To add or subtract, numbers must be in the same base.

In performing addition or subtraction,
whatever the base, the digits to be added or subtracted must be in the same place
value. For example in 65ten + 18ten, 5 and
8 have the same place value while 6 and 1 have another place value.
The base used is 8.
This is the required table

• The numerals used range from 0 to 20.

• Some examples of questions and answers
11 – 2 = 7; 17 – 10 = 7, 10 – 1 = 9 etc

Note:

• While working in base eight, eight
must not be one of the numerals in
use.

• In base eight, there are only 8 digits
i.e. 0, 1, 2, 3, 4, 5, 6, 7

Example 2.6

Evaluate: 332six + 25six

Solution

It is best to set work vertically so that the place values correspond.

                       +256
1. Illustrate the two numbers on
   different abaci (Fig. 2.9).

2. Remove all the 5 beads from R and
    place them in C to make 7 beads.One bead remains at C another
    goes to B to represent another group of six
    (Fig. 2.10).

3. Remove the two beads from Q and place them on B to make
  6 beads. No bead remains at B, but one bead goes to A to
 represent another group of six sixes.
 (Fig. 2.11).

4. The result of the addition is 401six
Alternatively,
332six → 330 + 2
25six → 20 + 5
            350 + 11
           = 350
           + 11/401six

Since we cannot subtract beads in R from
beads in C,

1. Remove one bead from B and place it on wire C so that there is a total
of 10 in C, (Fig. 2.13)

2. Remove 3 beads from C and R (Fig 2.14).

3. Remove 2 beads from B and Q so that the result is as represented in
       Fig. 2.14 below.
   ∴ 52₈ – 23₈ = 27₈

Alternatively,
 52₈ → 50 + 2 → 40 + 10
 – 23₈ → 20 + 3 → 20 + 3
                          20 + 7 = 27 _eight

Exercise 2.5

1. Work out the following in base eight:

(a) 17 + 211        (b) 106 + 12

(c) 257 + 462
2. Evaluate the following in base six:

(a) 31 – 25          (b) 145 – 51

(c) 55 – 43          (d) 403 – 54
3. Evaluate the following in base nine:

(a) 122 + 85        (b) 103 – 86

(c) 17 – 8             (d) 66 + 35

4. The following calculations are correct.
  State the base used in each case.

(a) 36       (b) 53           (c) 3
+ 26             + 36           +   23
–––––       –––––           –––––
64               111                31
–––––       –––––            –––––

5. Each of the following calculations
were done using a certain base. Three
of them are correct.
Identify:
(a) the base

(b) the incorrect ones and explain why.

(i) 22             (ii) 68
– 16               + 15
–––––            ––––
    6                    84

(iii) 100        (iv) 177
    – 64                  + 19
   –––––             ––––––
    25                     207

2.3.2 Multiplication

   Activity 2.8
Table 2.4 shows part of the
multiplication table for numerals in a certain base.

(a) Identify the base.

(b) Copy and complete the table.

(c) Given that x is a numeral, use your
    table to find the value of x if:
    (i) 6x = 13        (ii) 3x = 23

    (iii) 7x = 46
  (d) Use your table to formulate three
       equations using a variable of your
       choice.

Note that for any base;
(i) the highest numeral is always one less than the base and
(ii) the least is zero (0).
   Consider the product:
2_six × 3_six
Whether in base ten or base six, 2 by 3
 remains the same
  2 × 3 = 6_ten = 106

Example 2.8
Use long multiplication to evaluate
45six × 23 six.
Solution
   ×45
     23
   ––––
    1340
  + 223
   –––––––
    2003
   –––––––
(i) 1st row products
     5 × 2 = 10_ten = (14_six, we write 4 and  carry 1)
     2 × 4 = 8_ten (12six plus the 1 we carried )
             = 12_six + 1

           = 13six (we write 13_six )
    45 × 2 = 134six
(ii) 2^nd row products
3 × 5 = 15ten = 23six (We write 3 and
carry 2)
3 × 4 = 12ten = 20six (20 plus 2 we carried)
20_six + 2_six = 22_six
∴ 45 × 3 = 223_six.
Add the products in the 1st and 2nd rows
to get 2003_six

2.3.3 Division

Activity 2.9

(a) Given that a, b and c are numerals
    in base ten such that ab = c,
               express:
(i) a in terms of b and c.

(ii) b in terms of a and c.

(iii) Describe the operation

used to obtain the results
      above.
(b) Given that 2_six x 5six = 14_six,
   express:
(i) 2six in terms of 5_six and 14_six.

(ii) 5six in terms of 2_six and 14_six.
   What operation have you used
  to obtain your results?

(c) Make a multiplication table for base six and use it to confirm
   your findings in part (b) above.

(d) Use the table in (c) above
     to create more examples of division.

Now consider the example 23_six ÷ 5_six.
To do this, you ask yourself, 'by what
can I multiply 5_six to obtain 23_six?'
This question can be answered using the
multiplication table.

Example 2.9

Evaluate: 15_six ÷ 2_six

Solution

2 × 5 = 10_ten = 14_six

15_six ÷ 2_six = 5 Rem 1

We could also divide by first changing
the number to base 10, then change back to base 6.

15_six = (1 × 6) + 5 = 11

2 _six = 2_ten

15_six ÷ 2_six = 11_ten ÷ 2_ten = 5 Rem 1

5_six = 5_ten and 1_six = 1ten

15_six ÷ 2_six = 5_six Rem 1
But this is a long and an unnecessary
process.

Exercise 2.6
1. Copy and complete the multiplication
table in base eight and use it to answer
the questions below.

(a) 52₈ ÷ 7₈          (b) 43₈÷ 5₈

(c) 34₈ ÷ 7₈          (d) 20₈÷ 4₈

2. Evaluate the following:

(a) 15_six × 11_six

(b) 21₆ × 12₆

(c) 5₆ × 5₆

(d) 1 333₆ ÷ 35₆

3. (a) 2 122₄ ÷ 23₄
   (b) 100 122₄ ÷ 203₄

4. (a) 1 216₈ ÷ 3₈

    (b) 1 032₆ ÷ 4₆

2.4 Special bases

2.4.1 The binary system (base two) Base two

Activity 2.10

1. Write down all the digits used in the base 10 system.

2. Convert each of the digits in (a) to base 5.

3. Present your findings in a table similar to table 2.6.

A binary system is a number system that
uses only two digits 0 and 1. Numbers are expressed as powers of 2 instead of
powers of 10 as in the decimal system. Computers use binary notation, the two
digits corresponding to two switching position, on and off, in the individual
electronic devices in the logic circuits. Remember; in any base there is no
numeral equal to the base. Such a numeral always takes the form of 10.

Note: Just as in division in decimal
system, remember to put a zero in the
answer any time the divisor fails to divide.

Exercise 2.7

1. Evaluate:

(a) 1011₂ + 1101₂

(b) 10001₂ + 110011₂

(c) 11101₂ + 11₂ + 10101₂

(d) 1₂ + 11₂ + 1011₂+ 110011₂

2. Calculate:

(a) 10111₂ – 1101₂

(b) 11000₂ – 1110₂

c) 11111₂ – 10010₂

(d) 1010101₂ – 1111₂

3. Evaluate:

(a) 101₂ × 11₂

(b) 1111₂ × 1101₂

(c) 10101₂ × 11₁

(d) 1110₂ × 111₂

4. Evaluate:

(a) 101011₂ ÷ 11₂

(b) 11100101₂ ÷ 101₂

(c) 10001011₂ ÷ 1011₂

(d) 100010011₂÷ 101₂

(e) 100100001₂ ÷ 10₂

5. Find the prime factors of 10111002.

6. Convert the following to the binary system.
 (a) 18_ten            (b) 135_six

(c) 65_seven        (d) 35_eight

7. Convert 10110two to base four.

8. Evaluate the following giving your
answers in base two.
(a) 15_ten + 23_ten        (b) 35_ten – 12_ten

2.4.2 Base twelve (Duodecimal system)

Activity 2.11

1. Think of examples of items
    where we group in twelves.

2. Use your dictionary to find the meaning of the word dozen.
 

A system of numbers whose base is twelve is called duodecimal system. When
buying or selling in bulk, often, items are counted in groups of twelve i.e. dozens.
Earlier in the chapter, we saw that the numeral equivalent to the base is always
represented by 10. Therefore, in the case of base twelve, we have to define two
different variables to use in place of 10 and 11 to avoid confusion. Such substitutions
are necessary when working with any base greater than 10, i.e. base eleven, thirteen

etc. To be able to list the digits used in base twelve, let letter A represent 10, and
B represent 11. Thus, the digits in base twelve are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B.

2.5 Solving equations involving numbers in other bases

Activity 2.12

Solve the equations;
(a) (i) x – 6 = 10       (ii) x –(–3)= –5
     (iii) x÷4 + 3 = 5
given that you are working
  with base 10 system.
(b) Solve the equations in (a) above
     using base 6.
Now, consider the equations below.
     (i) x + 5 = 12              (ii) x – (–2)= 4
     (iii) 2x÷3 + 3 = 5
Working with base ten system,
      (i) x + 5 =12 ⇒ x = 7
      (ii) x + 2 = 4 ⇒ x = 2

(iii) 2x÷3 + 3 = 5 ⇒ x = 3
     Now, working in base six,
(i) x + 5 = 12 can be written as
    x + 5 = 20 (since 1210 = 206

  ⇒ x + 5 = 20
           x = 20 – 5
             = 11six
Alternatively, we can assume that

x + 5 = 12
x = 12 – 5
  = (6 + 2) – 5
  x = 3_six
(ii) x – (–2) = +4
     x + 2 = 4
     x = –2 + 4
       = 2six
(iii) 2x÷3 + 3 = 5…..3 × 3 = 910 =13six
                    And 5 × 3 = 1510 = 23six

Thus 2x÷3 + 3 = 5 becomes
2x + 13 = 23 (base six)…..
  multiplying each
 term by 3
2x = 23 – 13……….. subtracting
                          13_six from
                            both sides
     = 10_six
x = 3_six………… after dividing both
sides by 2_(six )

Example 2.13

Solve for the unknown in:
(a) 2x + 15 = 17 (base 8)

(b) 1÷2x – 3x = 20 (base 6)

(c) 3÷4x + 3÷2x = 9 (base 10)

Example 2.14

Solve for x in the equation 36_x + 26_x =64_x
given that x is a number other than base
ten. Verify your answer.

Solution

36_x + 26_x = 64_x can be expressed as
3 × x + 6 + 2 × x + 6 = 6 × x + 4
(convert the equation to base ten then
solve for x)
3x + 6 + 2x + 6 = 6x + 4
             5x + 12 = 6x + 4
            12 – 4 = 6x – 5x
                   8 = x
            Thus x = 8.
To verify the answer, substitute 8 = x in the
given equation 36x + 26x = 64x
In 36x + 26x = 64x,
LHS 36x + 26x = 3x + 6 + 2x + 6
                    = 3 × 8 + 6 + 2 × 8 + 6
                    = 24 + 6 + 12 + 6
                    = 30 + 22
                    = 52

RHS 64_x = 6x + 4
               = 6 × 8 + 4
              = 48 + 4
             = 52
Thus LHS = RHS = 52
x Represents base 8 in the given equation.
Exercise 2.9
1. Solve for x in:

(a) 2102₃ = 72_x

(b) 110011_two = 23_x

2. Solve for x if 110101₂ = x₈

3. Given that A and B represent ten and
eleven respectively in a certain base x,
solve for x in:

(a) A7_x + 5B_x = 198₁₀

(b) BA1_x = 1705₁₀

4. Find x if 10011_two = 23_x.

5. Given that x is the base, solve the
equation:

(a) 25_x + 13_x = 42_x

(b) 32_x + 24_x= 100_x

(c) 142_x + 33_x = 215_x

6. Solve for x in the following:

(a) 12_x – 6_x = 5_x

(b) 31_x – 16_x = 12_x

(c) 32_x – 24_x = 6_x

(d) 142x – 53_x = 67_x

7. Given that A and B represent 10 and
11 respectively in base twelve, solve
   for x in;
(a) 12A₁₂ + 4AB₁₂ = x_ten

(b) 789₁₂ – AB₁₂ = x _nine

Unit Summary

• A numeral is a symbol for a number
for example number twenty five is represented by the numeral 25.

• A numeral is composed of one or more digits. Thus a digit is a single numeral.

• 2 is a numeral mode of a single digit
   365 is a numeral, each of 3, 6, 5 is a digit 365 is made up of three digits.

• In base ten, we use nine digits or
  numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in base 8, we need eight digits i.e 0, 1,
   2, 3, 4 , 5, 6, 7 working with number bases, we never use a numeral equal
   to the base.

• The value of a digit depends on its
   position in the numeral thus, in a number such as
  452_ten the value of 4 is 4 × 10², the value of 5 is 5 × 10 And the value
  of 2 is simply 2.  452 = 4 × 10² + 55 × 10 + 2
   Similarly, in 452_six, the value of 4 is
  4 × 62, that of 5 is 5 × 6 and 2 remains 2.
∴ 452_six = 4 × 6² + 5 × 6 + 2,
               = 176_ten.
• If a base is greater than ten, the
  numerals above ten must be represented by a simple variable or
  symbol. For example, in base 12 we need to invent a symbol to represent
  10 and 11. We could use A for 10, B for 11 or any other variable provided
   we define it i.e we could say let
     A = 10
     B = 11
The number equal to the base is
always represented by 10.
In base 12, we use 10 for 12, in base
13 we use 10 for 13 and so on.

Unit 2 Test
1. Add 3554_six to 44_six giving your
answer in the same base.

2. (a) Express 101_eight in base 2.

   (b) Calculate 110_two × 1010_two giving
   your answer in base two and also
    in base ten.

3. Write 230_n as an algebraic expression
    in terms of n.

4. Given that 10022_three = 155_n, find the
  value of n.

5. Write each of the numbers as a mixed
number in the base number

(a) 101.11_ten

(b) 21.01_five

6. If 13 × 21 = 303 find the base of the
  multiplication.
7. In this binary addition, t and r, stands
   for a particular digit i.e 0 or 1 find t
   and r complete the addition.
1trrt
+ 1trr/1....r

8. Carry out the following in base six

(a) 115 + 251 + 251

(b) 53412 - 34125

(c) 123 × 54

9. If A stands for 10 and B stands
for eleven, perform the following
duodecimal calculations:

(a) 59A + AB          (b) 4A + AB + 9

(c) 10 × 54           (d) 45B – A1

(e) 11 × 7 – 8        (f) 32 + 6B

(g) 159A – 6BA

10. Solve the equation:

  31_x – 17_x = 16_x