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Real numbers

Commutative and Associative Properties of Rational Numbers

Concepts Related to Surds

Look at the following numbers.

All these are rational numbers as .

Now, observe the numbers . These numbers are irrational.

Roots of rational numbers:

Suppose 5 is the square of a rational number, then

x2 = 5

x =

Here, 5 is a rational number, but is not a rational number. Thus, x can not be a rational number.

Now, let us assume that 10 is the cube of a rational number, therefore y3 = 10.


Here, 10 is a rational number. Since cube root of 10 is not a rational number, y cannot be a rational number.

Similarly, there are many rational numbers that are not square, cube, etc. of any rational number. In other words, we can say that there are many rational numbers whose roots are irrational.

Irrational root of a positive rational number is called surd.

For example:

, , , , etc.

It can be generally defined in the following way:

If is an irrational number such that x is a positive rational number and a (a ≠ 1) is a natural number, then is known as a surd. Here, is the radical sign, a is the order of the surd and x is the radicand.

When a = 2, the surd is called a quadratic surd.

Now, consider the number .

Is it a surd?

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