Real Number System

Tanusree Das

Scottish Church College

B Ed in Mathematics

Roll no. T/14-120

INTRODUCTION OF REAL NUMBERS: ................................................... 3

DIFFERENT TYPES OF REAL NUMBERS: .................................................................... 4

PROOF THAT THE SQUARE ROOT OF 2 IS NOT RATIONAL: .................................... 5

NATURAL NUMBER: ................................................................................... 6

BASIC OPERATIONS: .................................................................................................... 6

SPECIAL NATURAL NUMBERS: ................................................................................... 7

WHOLE NUMBER: ....................................................................................... 8

INTEGER NUMBER: ..................................................................................... 8

RATIONAL NUMBER: ................................................................................. 9

WRITING RATIONAL NUMBERS: ................................................................................ 9

Fraction form: ......................................................................................................... 10

Terminating decimals: ........................................................................................ 10

Repeating decimals: ............................................................................................. 10

Arithmetic: ................................................................................................................ 11

IRRATIONAL NUMBERS: ........................................................................ 11

EXAMPLES OF IRRATIONAL NUMBERS: ................................................................. 12

NUMBER LINE: .......................................................................................... 13

ALGEBRAIC PROPERTIES SATISFIED BY THE REAL NUMBERS: 13

REAL NUMBER EXERCISES: ................................................................... 14

Introduction of Real Numbers:

A real number is a rational or irrational number. Usually when people say "number" they usually mean "real number". The official symbol for real numbers is a bold R or a blackboard bold .

Some real numbers are called positive. A positive number is "bigger than zero". You can think of the real numbers as an infinitely long ruler. There is a mark for zero and every other number, in order of size. Unlike a ruler, there are numbers below zero. These are called negative real numbers. Negative numbers are "smaller than zero". They are like a mirror image of the positive numbers, except they are given minus signs (–) so that they are labeled differently from the positive numbers.

There are infinitely many real numbers. There is no smallest or biggest real number. No matter how many real numbers are counted, there are always more which need to be counted. There are no empty spaces between real numbers. This means that if two different real numbers are taken, there will always be a third real number between them, no matter how close together the first two numbers are.

If a positive number is added to another positive number, that number gets bigger. Zero is also a real number. If zero is added to a number, that number does not change. If a negative number is added to another number, that number gets smaller.

The real numbers are uncountable. That means that there is no way to put all the real numbers into a sequence. Any sequence of real numbers will miss out a real

number, even if the sequence is infinite. This makes the real numbers special. Even though there are infinitely many real numbers and infinitely many integers, we can say that there are "more" real numbers than integers because the integers are countable and the real numbers are uncountable.

Some simpler number systems are inside the real numbers. For example, the rational numbers and integers are all in the real numbers. There are also more complicated number systems than the real numbers, such as the complex numbers. Every real number is a complex number, but not every complex number is a real number.

Different types of real numbers:

There are different types of real numbers. Sometimes all the real numbers are not talked about at once. Sometimes only special, smaller sets of them are talked about. These sets have special names. They are:

Natural numbers: These are real numbers that have no decimal and are bigger

than zero.

Whole numbers: These are positive real numbers that have no decimals, and

also zero. Natural numbers are also whole numbers.

Integers: These are real numbers that have no decimals. These include both

positive and negative numbers. Whole numbers are also integers.

Rational numbers: These are real numbers that can be written down as

fractions of integers. Integers are also rational numbers.

Irrational numbers: These are real numbers that can not be written as a

fraction of integers. Transcendental numbers are also irrational.

The number 0 (zero) is special. Sometimes it is taken as part of the subset to be considered, and at other times it is not. It is the Identity element for addition and subtraction. That means that adding or subtracting zero does not change the original number. For multiplication and division, the identity element is 1.

One real number that is not rational is . This number is irrational. If a square is drawn with sides that are one unit long, the length of the line between its

opposite corners will be .

Proof that the square root of 2 is not rational:

The number is not rational. Here is the proof.

1. Assume that is rational. So there are some numbers such

that .

2. We can choose a and b so that either a or b is odd. If a and b were both

even, then the fraction could be simplified (for example, instead of

writing , we could write instead).

3. If both sides of the equation are squared, then we get a2 / b2 = 2 and a2 =

2 b2.

4. The right side is . This number is even. So the left side must be even too.

So is even. If an odd number is squared, then an odd number will be the

result. And if an even number is squared, an even number would be the

result too. So is even.

5. Because a is even, it can be written as: .

6. The equation from the step 3 is used. We get 2b2 = (2k)2

7. An exponentiation rule can be used (see the article) – the result

is .

8. Both sides are divided by 2. So . This means that is even.

9. In step 2, we said that a is odd or b is odd. But in step 4, it was said that a is

even, and in step 7, it was said that b is even. If the assumption we made in

step 1 is true, then all these other things have to be true, but since they

disagree with each other they can not all be true; that means that our

assumption is not true.

It is not true that is a rational number. So is irrational.

Natural number:

Natural numbers, also called counting numbers, are the numbers used for counting things. Sometimes the special number zero is called a natural number. Sometimes one is called the smallest natural number. Natural numbers are always whole numbers (integers) and never less than zero.

There is no largest natural number. The next natural number can be found by adding 1 to the current natural number, producing numbers that go on "for ever". There is no infinite natural number. Any natural number can be reached by adding 1 enough times to the smallest natural number.

Basic operations:

Addition: The sum of two natural numbers is a natural number.

Multiplication: The product of two natural numbers is a natural

number.

Ordering: Of two natural numbers, if they are not the same, then one is bigger

than the other, and the other is smaller. m = n or m > n or m < n

if l > m then l + n > m + n and l x n > l x m

Zero is the smallest natural number: 0 = n or 0 < n

There is no largest natural number n < n + 1

Subtraction: If n is smaller than m then m minus n is a natural number. If n < m

then m - n = p.

if l - m = n then l = n + m

if n is greater than m, then m minus n is not a natural number

if l = m - n and p < n then l > m - p

Division:

Special natural numbers:

Even numbers: If n = m x 2, then n is an even number

The even numbers are 0, 2, 4, 6, and so on. Zero is the smallest (or first)

even number.

Odd numbers: If n = m x 2 +1, then n is an odd number

A number is either even or odd but not both

The odd numbers are 1, 3, 5, 7, and so on.

Composite numbers: If n = m x l, and m and l are not 0 or 1, then n is a

composite number.

The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, and so on.

Prime numbers: If a number is not 0, 1, and not a composite number, then it is

a prime number

The prime numbers are 2, 3, 5, 7, 11, 13, 17, and so on. Two is the smallest

(or first) prime number. Two is the only even prime number.

There is no biggest prime number.

Square numbers: If n = m x m, then n is a square. n is the square of m.

The squares are 0, 1, 4, 9, 16, 25, and so on.

Whole number:

These are positive real numbers that have no decimals, and also zero. Natural

numbers are also whole numbers.

Is Zero natural number? No, Zero is not a natural number.

Integer Number:

Integers are the natural numbers and their negatives.

These are some of the integers:

Zero is also an integer but it is neither positive nor negative. "Integer" is another word for "whole". An integer is a rational number with no "fraction", or part. An integer is a decimal number with all zeros after the decimal separator. (For example, the integer 17 is the same as the decimal 17.0 or 17.0000.)

An integer has a next smaller number and a next larger number. There is no smallest integer. There is no largest integer. Each integer is either larger than, equal to, or smaller than any integer. Consecutive integers are integers that come after each other, like

The sum of integers is an integer. The difference between integers is an integer. The product of integers is an integer. (For example, (12 + 2345 x (67 - 8)) x 9 is an integer.) An integer divided by an integer is sometimes not an integer. (For example, 123 / 45 is not a integer.)

Rational number:

In mathematics, a rational number is a number that can be written as a fraction. Rational numbers are all real numbers, and can be positive or negative. A number that is not rational is called irrational.

Writing rational numbers:

Fraction form:

All rational numbers can be written as a fraction. Take 1.5 as an example. This can

be written as , , or .

More examples of fractions that are rational numbers include , , and .

Terminating decimals:

A terminating decimal is a decimal with a certain number of digits to the right of

the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are

rational. Another good example would be 0.9582938472938498234.

Repeating decimals:

A repeating decimal is a decimal where there are infinitely many digits to the right

of the decimal point, but they follow a repeating pattern.

An example of this is . As a decimal, it is written as 0.3333333333... The dots tell

you that the number 3 repeats forever.

Sometimes, a group of digits repeats. An example is . As a decimal, it is written

as 0.09090909... In this example, the group of digits 09 repeats.

Also, sometimes the digits repeat after another group of digits. An example is . It

is written as 0.16666666... In this example, the digit 6 repeats, following the digit

1.

If you try this on your calculator, sometimes it may make a rounding error at the

end. For instance, your calculator may say that , even though

there is no 7. It rounds the 6 at the end up to 7.

Arithmetic:

Whenever you add or subtract two rational numbers, you always get another

rational number.

Whenever you multiply two rational numbers, you always get another rational

number.

Whenever you divide two rational numbers, you always get another rational

number, as long as you do not divide by zero.

Two rational numbers and are equal if .

Irrational numbers:

Irrational numbers are numbers which cannot be written as a fraction, but do not

have imaginary parts.

√2 is irrational.

Irrational numbers often occur in geometry. For instance if we have a square which has sides of 1 meter, the distance between opposite corners is the square root of two, which equals 1.414213 ... . This is an irrational number. Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.

One well-known irrational number is pi. This is the circumference (distance around) of a circle divided by its diameter(distance across). This number is the same for every circle. The number pi is approximately 3.1415926535 ... .

An irrational number cannot be fully written down in decimal form. It would have an infinite number of digits after the decimal point. Unlike 0.333333 ..., these digits would not repeat forever.

Examples of Irrational Numbers:

Rational (terminates)

Rational (repeats)

Rational (repeats)

Rational (repeats)

Irrational (never repeats or terminates)

Irrational (never repeats or terminates)

Number line:

In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point.

Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.

It is divided into two symmetric halves by the origin, that is the number zero.

In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.

Algebraic properties satisfied by the Real

numbers:

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

Closure: under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.

Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.

Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.

Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).

No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = any number.

Real Number Exercises:

Directions and/or Common Information: Classify these sets of terms.

4, -3, 12, 22, -4, 33

1. whole numbers

integers

rational numbers

irrational numbers

2. whole numbers

integers

rational numbers

irrational numbers

3. whole numbers

integers

rational numbers

irrational numbers

4,5,44,754,96

4. whole numbers

integers

rational numbers

irrational numbers