Basic Mathematics Form One Notes – Rational and Irrational Numbers

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Rational Numbers

A Rational Number

Define a rational number

A Rational Number is a real number that can be written as a simple fraction (ie. as a ratio),

Most numbers we use in everyday life are Rational Numbers.

Number Asa Fraction Rational?

5 sn Yes

175 74 Yes

oot 1/1000 Yes

1 “110 Yes

oun “9 Yes

V2ssquare root of 2) ? Not

The square root of 2 cannot be written as a simple fraction! And there are many more such

numbers, and because they arenot rationalthey are calledirrational.

The Basic Operations on Rational Numbers

Perform the basic operations on rational numbers

Addition of Rational Numbers:

To add two or morerational numbers, the denominator of all the rational numbers should be the

same. If the denominators of all rational numbers are same, then you can simply add all the

numerators and the denominator value will the same. If all the denominator values are not the

103

same, then you have to make the denominator value as same, by multiplying the numerator and

denominator value by a common factor.

Example 1

3443-53

13 +1915 +315 =815

Subtraction of Rational Numbers:

To subtract two or more rational numbers, the denominator of all the rational numbers should be

the same. If the denominators of all rational numbers are same, then you can simply subtract the

numerators and the denominator value will the same. If all the denominator values are not the

same, then you have to make the denominator value as same by multiplying the numerator and

denominator value by a common factor.

Example 2

BBW

V3-VS=915-15-215

Multiplication of Rational Numbers:

Multiplication of rational numbers is very easy. You should simply multiply all the numerators

and it will be the resulting numerator and multiply all the denominators and it will be the

resulting denominator.

Example 3

43×23-89

Division of Rational Numbers:

Division of rational numbers requires multiplication of rational numbers. If you are dividing two

rational numbers, then take the reciprocal of the second rational number and multiply it with the

first rational number.

Example 4

108

4B+25=43×52=206=103

Irrational Numbers

Define irrational numbers

An irrational number is areal numberthat cannot be reduced to any ratio between anintegerpand

anatural numberg. The union of the set of irrational numbers and the set of rational numbers

forms the set of real numbers. In mathematical expressions, unknown or unspecified irrationals

are usually represented byuthroughz. Irrational numbers are primarily of interest to theoreticians

Abstract mathematics has potentially far-reaching applications in communications and computer

science, especially in data encryption and security,

Examples of irrational numbers are V2 (the square root of 2), the cube root of 3, the circular ratio

, and the naturallogarithmbasee. The quantities\2 and the cube root of 3 are examples of

algebraic numbers. Pi and eare examples of special irrationals known as_atranscendental

numbers. The decimal expansion of an irrational number is always nonterminating (it never

ends) and non repeating (the digits display no repetitive pattern)

Real Numbers

Define real numbers

he type of number we normally use, such as 1, 15.82, -0.1, 3/4, ete Positive or negative, large or

small, whole numbers or decimal numbers are all Real Numbers.

They are called “Real Numbers” because they are not Imaginary Numbers.

Absolute Value of Real Numbers

Find absolute value of real numbers

The absolute value of a number is the magnitude of the number without regard to its sign. For

example, the absolute value of x or x written as. The sign before x is ignored. This is because

the distance represented is the same whether positive or negative. For example, a student walking

5 steps forward or 5 steps backwards will be considered to have moved the same distance from

where she originally was, regardless of the direction,

105

|<—5steps —»|<«——5steps—+

+4_1_1_1_1+_+_1+_+_+_1_++-»

54-32-0123 45

The 5 steps forward (+5) and 5 steps backward (-S) have an absolute value of S

Thus |x| = x when x is positive (x > 0), but |x| = x when x is negative (x <0)

For example, |3|=3 since 3 is positive (3 > 0) And ~3 = (3) =3 since ~3 is negative (3 < 0)