Rational and Irrational Numbers

Rational Numbers: 
     The rational number is expressed as frational form i.e. m/n are callled rational numbers. Here , m and n are integers but n is not equal to zero .
eg., 5/2 is a rational number it means that 5 is divided by 2 .

Comparison of Rational numbers :
 
      For any pair of number line the number to the lest is smaller that the other. Also , if the numeraror and the denominator of a rational number is multiplied by any non zero number then the value of rational number does not change. It remains the same . That is a/b = ka/kb , (k ≠ 0).
eg ., Compare the number 5/4 and 2/3 . write using the proper symbol of <, > , =.
solution :
    5/4  = 5 х 3 / 4 х 3  = 15/12
    2/3  = 2 х 4 /3 х  4 =  8/12 
 
  15/12  >  8/12    ∴  5/4  > 2/3  

Decimal Representation of  Rational Numbers :

     If we use decimal fractions which dividing the numerator of a rational numbe by its denominator , we get the decimal representation of a rational numbe . For example 5/2 = 2.5 .
  Such a decimal form of a rational number is called a terminating decimal form.
The every rational number can be written in a non-terminating recurring decimal form.
eg., i)   7/6 = 1.1666  =1.16
      ii)  5/6 = 0.8333 = 0.83
     iii) -5/3 = - 1.666 = -1.6 .
     iv) 22/7 = 3.142857142857... = 3.14257 
 
 similarly , a terminating decimal form can be written as a non-terminating recuring decimal form . For exampl, 7/4 = 1.75 = 1.75000... = 1.750 .

Standard Form of Rational Numbers 

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.

For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.

Arithmetic Operations on Rational Numbers

Addition: When we add p/q and s/t, we need to make the denominator the same. Hence, we get (pt+qs)/qt.

Example: 2/3 + 4/2 = 4+12/6 = 16/6

Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.

Example: 4/3 - 2/4 = 16-6/12 = 10/12 = 5/6

Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).

Example: 4/3 🇽 5/2 = 20/6 = 10/3

Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs

Example: 4/3 ➗ 5/2 =  4/3 🇽 5/2 = 20/6 = 10/3 

Irrational Numbers :

  The number which are not rational number is called as irrational number. eg.,  √2 is a irrational number .

Irrational numbers have endless non-repeating digits after the decimal point. Below is an example of an irrational number:

Example: √8 = 2.828…

Irrational Number Symbol

Generally, the symbol used to represent the irrational symbol is “P”.  Since irrational numbers are defined negatively, the set of real numbers (R) that are not the rational number (Q) is called an irrational number. The symbol P is often used because of the association with the real and rational number. (i.e.,) because of the alphabetic sequence P, Q, R. But mostly, it is represented using the set difference of the real minus rationals, in a way R- Q or R\Q.

Properties of Irrational numbers

 The following are the properties of irrational numbers:

  • The addition of an irrational number and a rational number gives an irrational number.  For example, let us assume that x is an irrational number, y is a rational number and the addition of both the numbers x +y gives an irrational number z.
  • Multiplication of any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational, then x =z/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.
  • The least common multiple (LCM) of any two irrational numbers may or may not exist.
  • The addition or the multiplication of two irrational numbers may be rational; for example, √2. √2 = 2. Here, √2 is an irrational number. If it is multiplied twice, then the final product obtained is a rational number. (i.e) 2.
  • The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.

Product of Two Irrational Numbers

Statement: The product of two irrational numbers is sometimes rational or irrational

For example, √4 is an irrational number, but when √4  is multiplied by √4, we get the result 4, which is a rational number.

(i.e.,) √4 x √4 = 4 

We know that π is also an irrational number, but if π is multiplied by π, the result is π2, which is also an irrational number.

(i.e..) π x π = π2

It should be noted that while multiplying the two irrational numbers, it may result in an irrational number or a rational number. 

Sum of Two Irrational Numbers

Statement: The sum of two irrational numbers may be rational or irrational.

Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.

For example, if we add two irrational numbers, say 3√2+ 4√3, a sum is an irrational number.

But, let us consider another example, (3+4√2) + (-4√2 ), the sum is 3, which is a rational number.

So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational number.


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