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MTE3101 Knowing Numbers

Topic 2Elementary Number Theory

2.1 Synopsis This topic covers different number systems and focusses on the definition of number systems, classifications within the set of real numbers and number representation. The number systems referred to in this topic concentrates on Real Numbers that includes the set of Natural Numbers, Whole Numbers, Integers, Rational Numbers and Irrational Numbers.2.2 Learning Outcomes

1. Generate one set of numbers to another set of numbers.

2. Characterise natural, rational, irrational and real numbers.

2.3 Conceptual Framework

2.4 Number SystemsNumber theory is one of the oldest branches of pure mathematics and focusses on the study of natural numbers. Arithmetic is taught in schools where children begin with learning numbers and number operations. The first set of numbers encountered by children is the set of counting numbers or natural numbers.In mathematics, a number system is a set of numbers. As mentioned earlier, children begin by studying the natural numbers: 1,2,3, ... with the four basic operations of addition, subtraction, multiplication and division. Later, whole numbers 0,1,2, .... are introduced, followed by integers including the negative numbers. The next step will include rational numbers and irrational numbers. In short, examples of number systems covered in this topic include natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers By studying the Number Systems it will help you to understand better the Elementary Number Theory in the next topic about Prime Numbers. Questions on divisibility, the use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, and number recreations such as Fibonacci numbers are included in the next topic which will be delivered to you face-to-face.2.4.1 DefinitionTo be good mathematics teachers, we need to possess a sound knowledge on different number systems at our fingertips. Knowing how to define the various sets of numbers within the real number system is therefore an absolute MUST!( Real Numbers

Lets begin by defining real numbers. Try and answer the following question. What is a real number?

A real number refers to any number that you would expect to find on the number line. It is a number whose name will be the "address" of a point on the number line. Its absolute value will name the distance of that point from 0. Real numbers contain all the rational numbers (which are the infinite repeating decimals, positive, negative and zero) together with a new set of numbers called the irrational numbers. In other words, the set of real numbers is the set of all numbers that have an infinite decimal representation.In schools, counting numbers are taught first, followed by whole numbers, fractions and integers. The relationship among these sets is illustrated below.

Each arrow represents is a subset of, for example, the set of counting numbers is a subset of the set of whole numbers, and so on. Subsequently, both the fractions and integers extend the system of whole numbers.The diagram in the previous page can be extended to include the set of rational numbers as follows:

Lets revise by considering the following definitions for the different sets of numbers summarised in the table below. The definitions are written using set notation. The { } symbols, called braces indicate the closing and opening of a set or collection of numbers. The three dots after the three indicate that the pattern continues. Definition of sets of numbersNamesSetsNotes and examples

Natural numbers{1, 2, 3, . . .} Represents all the counting numbers beginning with 1

Whole numbers{0, 1, 2 , 3, . . .} Starts with zero plus all natural numbers

Integers {0, 1, 2, 3,. . .} Includes negative, 0 and positive whole numbers

Rationalnumbers { | p and q are integers, q 0 } Read as a fraction p over q, where p and q are both integers, q 0 .Rational numbers can be written in decimal form, but they always either end or repeat (recur). Examples include:

Irrational numbers

{x | x is a nonrepeating and nonterminating decimal}Examples include:

pi () 3.14159. . , ; e 2.71828 ; 2 , etc.

Real numbers{x | x can be written as a decimal}

Read as all numbers x, such that x can be written as a decimal

2.4.2 Classifications within the set of real numbersIn mathematics, different types of numbers are grouped together and given names. It is important to understand this organization of sets of numbers. Real numbers can be classified under different sets of numbers. Look at the list of numbers in the table given in the previous page. What do you notice? Note that as you go down the list, a new set will contain the set of numbers directly above it. For example, the whole numbers contain the natural numbers. In fact, the set of whole numbers consists of all the natural numbers together with one new number, zero. As you go down the list, the numbers get more "complicated." Theprogression of numbers is much the way we learn about numbers as we grow up. As small children, we start with the natural numbers when counting our fingers and toys. We then make an intellectual leap and learn about the idea of "all gone" or no more left and the concept of zero, which takes us towhole numbers. Fractions were introduced because of the need to deal with parts of a whole. At some time in our development, we learn about debts and negative numbers, and we start using integers. The same sort of progression happens in mathematics classes. You start doing mathematics with whole numbers, then fractions as well as decimals followed by operations with negatives and positives. Notice that the integers are all members of the rational numbers. Any integer can be written as a rational number by writng a one under it. The only exception to this progression is the irrational numbers. They are by their own. The set of irrational numbers is the set of numbers that have infinite nonrepeating decimal representations. Thus, the rationals and the irrationals are disjoint sets. These two sets together make up the real numbers, that is, when we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number.From the above explanations on various sets within the real number system, you can now see how sets of numbers are related to one another and classified progressively. Now, can you describe the relationship between these sets?

Remember your Venn diagrams. The relationship between sets of numbers can be clearly shown with the help of Venn diagrams. The following diagram illustrates the relationships of the sets that make up the real numbers.

Next, something for you to think about.Testing your Understanding!1. Determine if the following statements are true or false. Give reasons for your answers.

i. Every integer is a rational number.

ii. Every rational number is an irrational number.

iii. Every natural number is an integer.

iv. Every integer is a natural number.2. Consider the following set of numbers:

{ - 81, - 0.315, 1, 3 , , 23, 6, 27, 3, 89.4, 100 000 }

Classify and list the numbers given according to the following sets. i natural numbers

ii whole numbers

iii integers

iv rational numbers

v irrational numbers

vi real numbers(Reminder: Dont forget to put your answers in your folio!)2.4.3 Number Representation

Besides using the set notation to represent various types of real numbers, we can also use other symbols such as alphabets to represent the set of real numbers. This is shown in the table below.Name of sets of numbersSymbols denoting the sets of numbers

Natural numbersN

Whole numbersW

IntegersZ

Rational NumbersQ

Irrational NumbersQ'

Real NumbersR

Apart from this, real numbers can also be represented on number lines. Writing numbers down on a number line makes it easy to tell which numbers are bigger or smaller. The ordered nature of the real numbers allow us to arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). The points are ordered so that points to the right are greater than points to the left, as shown in the diagram below.The Number Line

Negative Numbers (-)Positive Numbers (+)

(The line continues left and right forever.)

Numbers on the right are bigger than numbers on the left:

8 is greater than 5

1 is greater than -1

But notice that -8 is smaller than -5

The number line above shows that

Every real number corresponds to a distance on the number line, starting at the centre (zero).

Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.

The arrows on the end indicate that it keeps going forever in both directions.

For example, the number line below shows the set of Natural numbers:

Try representing the other sets of numbers discussed above using number lines. Have fun!

In conclusion, Real Numbers comprise the following: Rationals+Irrationals

All points on the number line

All possible distances on the number line.The discussion above serves to help you to recognise and cha