Equalities, inequalities and intervals

Once we have established the real line, we can compare a real number with another one.

If a number x lies to the right (left) of a number y, then we say that x is greater (smaller, respectively) than y, and express the relation as x > y (x < y, respectively). In contrast to these inequalities, if numbers x and z lie at exactly the same position, we say that x is equal to z, and we express the relation as x= z.

By combining these inequalities and equalities, we can describe intervals. For example, 0 < x < 1 implies that x can take any values between 0 and 1, not including either 0 or 1.

The expression x∈ (0, 1) carries the same information. The interval where neither of the end points is included is called an open interval. If both of the end points are included, then the interval is called a closed interval. For example, 0≤ x ≤ 1 implies that x can take

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2 Note that subtraction (denoted by−) and division (denoted by ÷) are their inverse operations, respectively.

13 2.5 Powers

any values between 0 and 1, including both 0 and 1; x∈ [0, 1] is an alternative expression for this interval (notice the difference in the type of brackets used). If 0≤ x < 1, then it implies that x can take any values between 0 and 1, including 0 but not including 1.

This is sometimes written as x ∈ [0, 1), and is called a half-open interval. Intervals that contain∞ are called infinite intervals. For example, to represent all the non-negative real numbers, we can write 0≤ x < ∞ or x ∈ [0, ∞). The set of all the real numbers can be written as−∞ < x < ∞ or x ∈ (−∞, ∞).

2.4.1 Absolute values

Now, it follows from the definitions of inequalities that−100 < 1 (it seems rather obvious, but try explaining why it is so). However, if we were to measure the distance from the centre of the real line, zero, clearly−100 is further away from zero than 1 is. In other words, if we ignore the minus sign,−100 has a greater magnitude than 1 does. We use an absolute value to show the magnitude of a number a, which is written as|a|. In the above example, |−100| > |1|. The interval −2 < a < 2 can be expressed using the absolute value as|a| < 2.

2.5 Powers

You will encounter the power of numbers in mathematics everywhere. Let us begin with the definition of a power. If we multiply a number q by the same number, the expression will be q× q. Another way of expressing it is q², where the superscript 2 is a power showing that two qs are to be multiplied together. In a sense, the power expression is like slang in a language; expressing the same thing in a different way. Just like you need to know slang to be an expert in a particular language, you need to know the power expression if you want to communicate well with others using mathematics.

In any case, let us go through the following process to see if we can learn something about the power expression. We know that multiplying a number by unity yield that number itself, so let us express q²as “a 1 multiplied by q twice,”

q² ≡ 1 × q × q.

By the same token, we can define q to various powers:

q³ ≡ 1 × q × q × q, q¹ ≡ 1 × q,

q⁰ ≡ 1.

Two things are worth noting. First, q is actually q¹but the superscript 1 is omitted for simplicity. Second, whatever the value q takes, we find that q⁰= 1.

We can extend this analysis to division and find out what negative powers look like.

Namely, for non-zero q:

q⁻¹ ≡ 1 ÷ q = 1 q, q⁻² ≡ 1 ÷ q ÷ q = 1

q²,

and so,

q^−m = 1 q^m. Now we know that q^−mis a reciprocal of q^m. 2.5.1 The basic power rule

There is only one basic power rule you need to remember because other rules follow this one. The important rule is:

q^m· qⁿ= q^m+n. (2.1)

Question Simplify q⁴· q³. Solution

q⁴· q³ = (q · q · q · q) · (q · q · q)

= q · q · q · q · q · q · q

= q⁴⁺³

= q⁷.

Exercise 2.2 Applying the power rule (2.1).

Now, using rule (2.1) we can deduce the following:

q^m· q⁻ⁿ= q^m+(−n) = q^m−n. Note that q⁻ⁿ= 1

qⁿ. Together, we have the following:

q^m÷ qⁿ= q^m

qⁿ = q^m−n. (2.2)

Question Simplify q⁶÷ q³. Solution

q⁶÷ q³ = q⁶ q³

= q· q · q · q · q · q q· q · q

= q⁶⁻³

= q³. Exercise 2.3 Applying the power rule (2.2).

15 2.5 Powers

Suppose now that y = q⁴. What is y³ in terms of q? This can be solved by directly applying the definition of the power:

y³ =

2.5.2 Non-integer powers, particularly 1 2

Up to this point, m and n were implicitly treated as integers and hence numbers were raised to some integers. But numbers need not be raised to integers only and the power rules can be applied for non-integers as well. To delve into this issue, let us look at the following statement: qⁿ= y. It follows from rule (2.3) that:

qⁿ = y^1n×n=

The above identity says that ‘q is the nth root of y’.

There are two special (or frequently used) ns. When n= 2, we say ‘q is the square root of y’. It follows from the convention that q² is called q-squared. As you may know, we can simply write it as q = √y (instead of√²

y). In turn, when n= 3, we say ‘q is the cube (or cubic) root of y’. It again follows from the convention that q³is called q-cubed.

Two remarks should be made regarding this introduction of non-integer powers. First, the number defined by a statement may not be unique. For example, 9 can be written as 9= 3 × 3 = 3² as well as 9= (−3) × (−3) = (−3)². So the statement q² = 9 can mean either q = 3 or q = −3. In general, we write it as q = ±3. Second, if we consider a statement such as q²= −16, we realise that there exists no ‘real’ number q that satisfies the statement. Mathematicians get away with this problem by ‘imagining’ that such numbers exist, and we introduce these imaginary numbers in the next section. But before that, let me discuss a little about some conventions regarding the square root.

2.5.3 Some conventions on the square root

There are two conventions concerning the square root that I want you to follow. Using a language metaphor, I would say that your mathematics will become more fluent by following these conventions.

The first convention is the following. When we end up with an expression with a square root of ‘a number multiplied by a squared number’, simplify the expression so that the number inside the squared root cannot be expressed as ‘a number multiplied by a squared number’. For example, do not leave the expression√

4 as it is. It be expressed as√ 2²and hence can be simplified to 2. A little more complicated example is√

120. You should not leave it as it is. This number can be (and should be) simplified to 2√

30.

The second convention is relevant when you end up with a fraction and the denominator includes a square root. If it occurs, do not leave the square root on the denominator. It is considered fine (or fluent, if you like) to have squared roots on the numerator, but not on the denominator. For example, when you end up with 1

√2, then do not leave it. Multiplying both the numerator and the denominator by the same number will not change the value of the expression, so let us use√

2 and see what occurs. The expression has now become