Key unit competence
Think critically using mathematical logic to understand and perform operations on the set of real numbers and its subsets using the properties of algebraic structures.
Work in groups.
1. Discuss the following: What are sets? What are real numbers?
2. Carry out research on sets of numbers to determine the meanings of natural numbers, integers, rational numbers and irrational numbers.
The one-to-one correspondence between the real numbers and the points on the number line is familiar to us all. Corresponding to each real number there is exactly one point on the line: corresponding to each point on the line there is exactly one real number.
4.2 Properties of real numbers
Subsets of real numbers
If a and b are any two real numbers, then either a < b or b < a or a = b. The sum and product of any two positive real numbers are both positive.
If a < 0 then (– a) >0.
If a < b then (a – b) > 0. This means that the point on the number line corresponding to a is to the right of the point corresponding to b.
The elementary rules for inequalities are:
1. If a < b and c is any real number then a + c < b + c . That is we may add (or subtract) any real number to (or from) both sides of an inequality.
2. (a) If a < b and c > 0, then ac < bc. (b) If a < b and c < 0, then ac > bc. We may multiply (or divide) both sides of an inequality by a positive real number, but when we multiply (or divide) both sides of an inequality by a negative real number we must change the direction of the inequality sign.
3. (a) If 0 < a < b then 0 < a2 < b2.
(b) If a < b < 0 then a2 > b2. We may square both sides of an inequality if both sides are positive. However if both sides are negative we may square both sides but we must reverse the direction of the inequality sign and then both sides become positive. If one side is positive and the other negative we cannot use the same rule. For example, –2 < 3 and (–2)2 < 32 , but –3 < 2 and (–3)2 > 22.
That is we may take the reciprocal of both sides of an inequality only if both sides have the same sign and, in each possible case, we must reverse the direction of the inequality sign.
4.3 Absolute value functions
In groups of four, research on the meaning of absolute value. Present your findings to the rest of the class for discussion. Make use of diagrams and examples.
There is a technical definition for absolute value, but you could easily never need it. For now, you should view the absolute value of a number as its distance from zero.
Let us look at the number line.
The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?" This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.
Note: The absolute-value notation is bars, not parentheses or brackets. Use the proper notation. The other notations do not mean the same thing.
As this illustrates, if you take the negative of an absolute value, you will get a negative number for your answer.
Sometimes we order numbers according to their size. We denote the size or absolute value of the real numbers x by |x|, called the modulus of x.
4.4 Powers and radicals
1. In pairs, revise the meaning of the term 'power'. Give the definition.
2. Research and derive the rules of exponential expressions.
The power of a number says how many times we use the number in a multiplication. It is written as a small number to the right and above the base number. Another name for power is index or exponent.
When a real number a is raised to the index m to give am , the result is a power of a. When am is formed, m is sometimes called the power, but it is more correctly called the index to which a is raised.
The rules used to manipulate exponential expressions should already be familiar with the reader. These rules may be summarized as follows:
Research on the meaning of the term radical. Discuss your findings with the rest of the class. Your teacher will assist you get a concise meaning of the term.
2. Every positive number has two square roots, one of which is positive and the other negative.
3. The square root of zero is zero.
4. Negative real numbers have no real square roots.
If a3 = b, then b is the cube of a and a is the cube root of b. This time we say the cube root of b since 23 = 8 but (–2)3 = –8.
1. No definition is needed to specify which cube root is meant since a real number has one and only one cube root in the real number system.
2. Every positive number has one and only one cube root which is positive.
3. The cube root of zero is zero.
4. Every negative number has one and only one cube root which is negative.
In general, if n is an even positive integer then in the real number system:
1. If a > 0, a has two nth roots, one positive and another negative.
2. If a = 0, a has one nth root which is zero.
3. If a < 0, a has no real nth roots.
If n is an odd positive integer (not 1) then in the real number system,
1. If a > 0, a has one nth root which is positive.
2. If a = 0, a has one nth root which is zero.
3. If a < 0, a has one nth root which is negative.
The process of squaring both sides is used in solving irrational equations, i.e., equations involving surd expressions. It is therefore always necessary to check any solution obtained.
4.5 Decimal logarithms
Discuss in groups of four and give two examples for each.
1. What is a decimal?
2. What is a logarithm?
The logarithm of a positive number, in base 10, is its power of 10. For instance:
Note: If no base is indicated we assume that it is base 10.
The laws of logarithms
Your teacher will guide you in deriving the laws of logarithms. Use numerous examples to practise using them.
The logarithm can be used in solving exponential equations.
Application of exponents in real life