IRRATIONAL NUMBERS

 Irrational Numbers

• Irrational numbers are those real numbers that cannot be represented in the form of a ratio. 
• In other words, those real numbers that are not rational numbers are known as irrational numbers

Irrational Numbers Definition

• An irrational number is a real number that cannot be expressed as a ratio of integers.
• for example, √ 2 is an irrational number. Again, the decimal expansion of an irrational number is neither terminating nor recurring.
• In other words, we can say that irrational numbers cannot be represented as the ratio of two integers.

Are Irrational Numbers Real Numbers?

• In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers. 
• It means that irrational numbers cannot be expressed as the ratio of two numbers. 
• The irrational numbers can be expressed in the form of non-terminating fractions and in different ways.
•  For example, the square roots which are not perfect squares will always result in an irrational number. 

Sum and Product of Two Irrational Numbers

Now, let us discuss the sum and the product of the irrational numbers.

Product of Two Irrational Numbers

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

For example, √2 is an irrational number, but when √2  is multiplied by √2, we get the result 2, which is a rational number.

(i.e.,) √2 x √2 = 2 

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 is sometimes 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.

Are Irrational Numbers Real Numbers?

In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers. It means that irrational numbers cannot be expressed as the ratio of two numbers. The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number. 

Irrational Numbers Symbols

Before knowing the symbol of irrational numbers, let us discuss the symbols used for other types of numbers.

• N- Natural Numbers
• I- Imaginary numbers
• R- Real Numbers
• Q- Rational Numbers

Properties of Irrational Numbers

• When an irrational and a rational number are added, the result or their sum is an irrational number only. For an irrational number x, and a rational number y, their result, x+y = an irrational number.
•   When any irrational numbers multiplied by any nonzero rational number, their product is an irrational number. For an irrational number x and a rational number y, their product xy = irrational.
•   For any two irrational numbers, their least common multiple (LCM) may or may not exist.
•  Addition, Subtraction, Multiplication and division of two irrational numbers may or may not be a rational numbers.

FUN FACTS

• Math lovers celebrate today (3/14) as Pi Day, in honor of the irrational number pi.
•  Pi, or π, is defined as the ratio of the circumference of a circle to its diameter. 
• Pi is an irrational number, meaning it cannot be written as a simple fraction.