UNIT 1A  NUMBERS

Exercises:   [1]   [2]   [3]

INTRODUCTION

This unit introduces you to some of the vocabulary, notation and basic tools of mathematics. An understanding of some of the difficulties encountered and overcome in the historical development of mathematical techniques may help you to appreciate the need for preceding in a precise and logical manner. If you do all the exercises faithfully you should develop some mathematical dexterity.

OBJECTIVES

1. To distinguish between natural numbers, integers, rational numbers and real numbers.
2. To evaluate arithmetic expressions.
3. To manipulate algebraic expressions.
4. To solve linear equations involving a single unknown.

REFERENCES

A popular discussion of number systems is given by Hogben in Mathematics for the million [7]. This is referred to specifically in the section on irrational numbers. Some interesting facts about numbers can be found in the Guinness book of records [6].

At the most basic level, mathematics is about numbers so why not start there?

NATURAL NUMBERS

These are the positive whole numbers used for counting: 1, 2, 3, 4, 5,  . Fingers ( and toes) are commonly used counters of natural numbers.

If you add two natural numbers you get another one, e.g. 1 + 2 = 3. Also if you multiply two of them you get another, e.g. 1 ´ 2 = 1 or 2 ´ 5 = 10. However if you subtract two natural numbers you may get a natural number,

e.g. 3  2 = 1

but you may not get a natural number,

e.g. 2  3 = ?

Similarly, if you divide two natural numbers you may get another natural number but you may not,

e.g. 6 ¸ 3 = 2 but 7 ¸ 2 = ?

To handle these situations which may occur in subtraction or division, more numbers had to be invented.

INTEGERS

These consist of the positive whole numbers, the negative whole numbers and zero:  -4, -3 ,-2, -1, 0, 1, 2, 3,  . You can add, multiply and subtract integers and you answer will always be another integer,

e.g. 5 + 3 = 8, 5  12 = -7, 2 ´ 3 = 6.

But you cannot always divide one integer by another and get an answer which is an integer,

e.g. 7 ¸ 2 = ?

USE OF BRACKETS

Brackets (or parentheses) are used to separate some part of a calculation from the rest. For instance, (10 ´ 3) + 2 means multiply 10 by 3 then add 2 to the product. What happens if the brackets are left out? For example, in

10 ´ 3 + 2

if you do the multiplication first you get the answer 32; on the other hand if you do the addition first the answer is 50.

The convention is that when brackets are omitted, multiplication and division are carried out first and then any addition or subtraction. Thus the expression 10 ´ 3 + 2 means (10 ´ 3) + 2, so the correct answer is 32. If there is a series of brackets nested within one another the expression in the innermost brackets is calculated first.

Arithmetic is the name given to the operations of adding, subtracting, multiplying and dividing numbers. If you have not been doing much arithmetic lately a few hints about the use of negative integers may be useful.

ARITHEMETIC INVOLVING NEGATIVE INTEGERS

Multiplication

When multiplying two negative integers remember that two negatives make a positive,

e.g. (-2) ´ (-3) = 6.

2 ´ (-3) = -6

(-1) ´ (-2) ´ (-3) = {(-1) ´ (-2)} ´ (-3)

= 2 ´ (-3) = -6.

e.g. 4 + (-3) = 1.

When a negative integer is added, subtraction notation is often used: i.e. 4 + (-3) is written as 4  3 but the meaning is still positive 4 add negative 3 gives positive 1,

e.g. (-2) + (-3) = -5.

Subtraction

This can always be written as addition,

e.g. 3  2 can be written as 3 + (-2) = 1

or 3 + (-1) ´ (2) = 1.

e.g. -5  3 = -5 + (-1) ´ (3) = -5 + (-3) = -8.

2  (-3) = 2 + (-1) ´ (-3) = 2 + 3 = 5.

FACTORS

If several numbers can be multiplied together to give another number they are called its factors.

e.g. 6 = 2 ´ 3 so 2 and 3 are factors of 6.

120 = 12 ´ 10 so 12 and 10 are factors of 120,

but

120 = 3 ´ 4 ´ 2 ´ 5 so 2, 3, 4, and 5 are also factors of 120.

A prime number is one that cannot be broken into factors; thus 1, 2 and 3 are prime numbers,

4 is not a prime number because 4 = 2 ´ 2,

5 is not a prime number,

6 is not (because 6 = 2 ´ 3), and so on.

e.g. 120 = 12 ´ 10 so 12 and 10 are factors of 120,

but

12 = 2 ´ 6 = 2 ´ 2 ´ 3 and 10 = 2 ´ 5 so

120 = 2 ´ 2 ´ 3 ´ 2 ´ 5 so the prime factors of 120 are 2 (three times over), 3 and 5.

Exercises 1.1 Solutions

1. Simplify:

(a) (-5) + 3

(b) (-8)  4

(c) (2 ´ 6) + (3 ´ 8)

(d) (-2) ´ (-3) ´ (-40

(e) (2) ´ (-3) ´ (-4)

(f) 9  3[6  2(4+1)  3(1-3)]

2. Find factors of the following integers:

(a) 100

(b) 58

(c) 37

(d) -6

RATIONAL NUMBERS

These are the positive fractions, negative fractions and zero. Any number of the form a ¸ b (also written as or a/b) is a rational number if a is any integer and b is any integer except zero.

e.g. , 2 ¸ 5 = , -1¸ 3 = = , are rational numbers, so are 4 ¸ 2 = and 5 ¸ 1 = (which is usually written simply as 5)

In fact any integer is also a rational number.

You can add, subtract, multiply or divide (except by zero) any two rational numbers and your answer will be another rational numbers.

SIMPLIFYING RATIONAL NUMBERS (CANCELLATION)

In the rational number p/q, p is called the numerator and q the denominator. If p and q share a common factor, the number p/q can be reduced to a simpler form by cancelling the factor from both p and q.

e.g. = =

e.g. 4 ¸ 2 = = = 2.

ARITHMETIC INVOLVING RATIONAL NUMBERS (EXCEPT ZERO)

Fluency with fractions is fundamental.

It is important that you develop a logical, systematic approach to the arithmetic of fractions because the principles involved apply more generally to many algebraic problems.

You cannot add 2 apples and 3 oranges. You need to think of them in terms of their common property fruit. Then, adding them, you have 5 pieces of fruit. The addition of fractions is analogous.

Example 1a. Simplify .

Step 1: Find the common denominator of and by multiplying their denominators 2 ´ 3 = 6.

Step 2: Write each fraction in terms of the common denominator

and .

Step 3: Add them, 3sixths plus 4 sixths give 7 sixths

.

Subtraction involves the same basic steps.

Example 1b. Simplify .

The common denominator is 3 ´ 6 = 18, and ,

so .

Multiplication

means "half of 4, i.e. 2. Similarly, means half of , i.e. . The method of calculation is to multiply the two numerators, then to multiply the two denominators and finally to cancel any common factors.

e.g. is the same as ,

.

e.g.

Example 1c Simplify .

.

Division

First consider division involving only integers, for example 4 ¸ 2 = 2. Dividing by 2 is equivalent to multiplying by , 4 ¸ 2 = 4 ´ = = 2.

e.g. 4 ¸ 3 = .

(-60)¸ 3 = .

In the above examples, both integers were converted to rational numbers (e.g. 4 =), then the divisor (the second number) was inverted and two numbers were multiplied.

Similarly, division by a fraction is equivalent to multiplication by the inverse of the fraction.

Example 1d. Simplify .

is rewritten as and then simplified,

.

Example 1e. Simplify .

Remember that multiplication (or division) is carried out before subtraction (or addition):

.

When multiplying or dividing negative numbers remember

TWO NEGATIVES MAKE A POSITIVE.

ARITHMETIC INVOLVING ZERO

If you add zero to or subtract it from any other number that number does not change,

e.g. 2 + 0 = 2, 6  0 = 6, 0 + ½ = ½ .

If you multiply by zero, the answer is always zero,

e.g. 5 ´ 0 = 0, 10000 ´ 0 = 0,

0 ´ = 0, (-6) ´ 0 = 0.

If zero is divided by another number the answer is zero,

e.g. 0 ¸ 6 = = 0.

You can NEVER DIVIDE BY ZERO

e.g. and are all undefined.

Exercises 1.2 Solutions

Simplify the following:

1.

2.

3.

4.

5. 6 + 5 ¸ 3

6. (21 + 14) / (8  6)

7.

8. (1 + )  2{6  0( + 1) + 2(1 - )}

POWERS

The notation used when a number is multiplied by itself is shown by the following examples:

36 = 6 ´ 6 = 62 (62 is read as 6 squared);

1000 = 10 ´ 10 ´ 10 = 103 (103 is read as 10 cubed);

5 = 51

, is read as  to the power 4 .

When numbers are multiplied, their powers (superscripts) are added,

e.g. 16 ´ 8 = (2 ´ 2 ´ 2 ´ 2) ´ (2 ´ 2 ´ 2)

= 24 ´ 23

= 24+3

= 27 i.e. there are 7 twos multiplied together,

(check that 16 ´ 8 = 128 and 27 = 2 ´ 2 ´ 2 ´ 2 ´ 2 ´ 2 ´ 2 = 128).

e.g. 6 ´ 36 = 61 ´ 62 = 61+2 = 63 = 216.

BE CAREFUL. You cannot add the powers in 25 ´ 35 because the numbers (bases) 2 and 3 are not the same.

The power of a number can be any integer: the meaning given to positive powers, zero powers and negative powers can be seen from the following pattern:

1000 = 103

100 = 102

10 = 101

1 = 100

= 10-1

= 10-2

= 10-3.

Any number raised to the power 1 remains unchanged,

e.g. 101 = 10, 61 = 6, 10001 = 1000.

Any number raised to the power 0 becomes 1,

e.g. 100 = 1, 50 = 1, 10000 = 1, = 1.

When numbers are divide, their powers are subtracted,

e.g. 3 ¸ 9 = 31 ¸ 32 = 31-2 = 3-1 = ,

(compare this with 3 ¸ 9 = ).

As you have seen, a power (also called an exponent), can take any integer value. A power may also be a rational number but, in order to avoid complications some restrictions the definition are needed. The number

ab

is defined when b is a (non-integer) rational number only if a is positive; also the value of ab is always taken to be positive. The meaning of rational powers is shown by the following examples.

e.g. 4½ = 2 because 4½ ´ 4½ = 4½+½ = 41 = 4 and

2 ´ 2 = 4;

4½ is called the square root of 4 and is also denoted by .

e.g. 271/3 = 3 because 3 ´ 3 ´ 3 = 33 = 27.

e.g. 272/3 = 271/3 ´ 271/3= (271/3)2 = 32 = 9.

Notice that by this definition of ab, = (-3)½ is undefined because a = -3 is not positive; also although (-2) ´ (-2) = 4, you cannot say that 4½ = -2 (because 2 is not positive).

Exercises 1.3 Solutions

1. Simplify the following expressions:

(a) 52 Χ 52 (b) 85 Χ 8-4 Χ 80

(c) 34 Χ 6-3 Χ 32 (d) {4 + (2 Χ 3)2}2

2. Evaluate:

(a) (b) 25

(c) 253/2 (d) 8-2/3

(e) 21/2 Χ 23/2 (f) (5½)2

(g)

DECIMALS

Using long division you can represent any rational number as a decimal. The decimal form may have only a finite umber of digits (other than 0) after the decimal points;

e.g. = 0.5, = 0.75,

or it may have infinitely many digits,

e.g. = 0.333, = 1.142857 142857 142 .

It can be proved that the representation of any rational number is either finite or, after a certain stage, the digits repeat themselves in groups.

IRRATIONAL NUMBERS

These are numbers which, when represented in decimal form, have infinitely many non-zero digits after the decimal point and the digits are not repeated systematically. Numbers with this property are not rational numbers but they occur in everyday life.

e.g. = 2½ = 1.414213 .

(The simple proof that is not rational can be found in many books, for example Mathematics for the million by L. Hogben [7].)

Another irrational number is

= 3.14159265 .

Here is a quotation from the Guinness book of records [6]: The greatest number of decimal places to which pi(p ) has been calculated is 1,000,000 by the French mathematicians Jean Guilloud and Mlle. Martine Bouyer achieved on 24 May 1973 on a CDC 7600 computer but not verified until 3 September 1973. The published value to a million places, in what has been described as the worlds most boring 200 page book was 3.141592653589793 (omitting the next 999,975 places)  5779458151. In 1897 the State legislature of Indiana came within a single vote of declaring that pi should be de jure 3.2.

PLEASE NOTE THAT p DOES NOT EQUAL ;

is a rational number which recurs after the sixth decimal places,

= 3.1428571428; it is only the same as p as far as the second decimal place.

REAL NUMBERS

These are all the rational numbers together with all the irrational numbers. You might guess that the real numbers would be all that you would ever encounter but this is not so. For example, what is the square root of (-2)? There is no real number which multiplied by itself gives the answer 2. Still more numbers, the complex numbers, have been invented to cope with the problem of finding roots of negative numbers. However, you will be relieved to know that throughout this course only real numbers are used.