Surds

We will discuss here about surds and its definition.

First let us recall about rational number and irrational number.

Before defining surds, we will first define what are rational and irrational number?

Rational number: A number of the form p/q, where p (may be a positive or negative integer or zero) and q (taken as a positive integer) are integers prime to each other and q not equal to zero is called a rational number or commensurable quantity.

Rational numbers are the numbers which can be expressed in the form of p/q where p is a positive or negative integer or zero and q is a positive or negative integer but not equal to zero.

Like: \(\frac{5}{7}\), 3, - \(\frac{2}{3}\) are the examples of rational numbers.

For example, each of the numbers 7, \(\frac{3}{5}\), 0.73, √25 etc. is a rational number. Evidently, the number 0 (zero) is a rational number.


Irrational number: A number which cannot be expressed in the form p/q where p and q are integers and q ≠ 0, is called an irrational number or incommensurable quantity.

Irrational numbers are the numbers which can’t be expressed in the form of p/q where p and q are integers and q ≠ 0. Irrational numbers have infinite numbers of decimals of non-recurring nature.

Like: π, √2, √5 are the irrational numbers.

For example, each of the numbers √7, ∛3, \(\sqrt[5]{13}\) etc. is an irrational number.


Definitions of surd: A root of a positive real quantity is called a surd if its value cannot be exactly determined.

Surds are the irrational numbers which are roots of positive integers and the value of roots can’t be determined. Surds have infinite non-recurring decimals. Examples are √2, √5, 17 which are square roots or cube roots or nth root of any positive integer.

For example, each of the quantities √3, ∛7, ∜19, (16)^\(\frac{2}{5}\) etc. is a surd.

From the definition it is evident that a surd is an incommensurable quantity, although its value can be determined to any degree of accuracy. It should be noted that quantities √9, ∛64, ∜(256/625) etc. expressed in the form of surds  are commensurable quantities and are not surds (since √9 = 3, ∛64 = 4, ∜(256/625) = \(\frac{4}{5}\) etc.). In fact, any root of an algebraic expression is regarded as a surd.

Thus, each of √m, ∛n, \(\sqrt[5]{x^{2}}\) etc. may be regarded as a surd when the value of m ( or n or x) is not given. Note that √m = 8 when m = 64; hence, in this case √m does not represent a surd. Thus, √m does not represent surd for all values of m.

8 or 81 can be simplified into 2 or 3 which are rational numbers or positive integers, 8 or 81 are not surds. But the value of √2 is 1.41421356…., so the decimals continue up to infinite numbers and non-recurring in nature, so √2 is a surd. π and e have also values which contain decimals up to infinite numbers but they are not root of positive integers so they are irrational numbers but not surds. So all surds are irrational numbers but all irrational numbers are not surds.

If x is a positive integer with nth root, then \(\sqrt[n]{x}\) is a surd of nth order when the value of \(\sqrt[n]{x}\) is irrational. In \(\sqrt[n]{x}\) expression n is the order of surd and x is called as radicand.

The reason that we leave surds in root form as the values can’t be simplified, so during problem solving with surds, we normally try to convert the surds to more simplified forms and whenever necessary we can take approximate value of any surd up to any decimal to calculate.  

Note: All surds are irrationals but all irrational numbers are not surds. Irrational numbers like π and e, which are not the roots of algebraic expressions, are not surds.

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Now we solve some problems on surds to understand more on surds.

1. Express √2 as a surd of order 4.

Solution

√2 = 2\(^{\frac{1}{2}}\)

     =2\(^{\frac{1 × 2}{2 × 2}}\)

     = 2\(^{\frac{2}{4}}\)

     = 4\(^{\frac{1}{4}}\)

     = \(\sqrt[4]{4}\)

\(\sqrt[4]{4}\) is a surd of order 4.


2. Find which are surds from the following numbers?

√24, 64 x √121, √50

Solution:

√24 = \(\sqrt{4 × 6}\)

       = 2√2 × √3

So √24 is a surd.

64 × √121 = \(\sqrt[3]{4^{3}}\) × √112

                  = 4 × 11

                  = 44

So 64 x √121 is rational and not a surd.

√50 = \(\sqrt{2 × 25}\)

       = \(\sqrt{2 × 5^{2}}\)

       = 5√2

So √50 is a surd.

If the denominator of a expression is a surd, then often it requires to convert the denominator to rational number. This process is called rationalizing or rationalization of surd. This can be done by multiplying a suitable factor to the denominator to convert the expression in to a more simplified form. This factor is called as rationalizing factor. If the product of two surds is a rational number, then each surd is a rationalizing factor to the other surd.

For example \(\frac{1}{2 + \sqrt{3}}\) is expression, where the denominator is a surd.

\(\frac{1}{2 + \sqrt{3}}\)

 = \(\frac{1\times (2 - \sqrt{3})}{(2 + \sqrt{3})\times (2 - \sqrt{3})}\)

\(\frac{(2 - \sqrt{3})}{4 - 3}\)

= 2 - √3

So the rationalizing factor of (2 + √3) is (2 - √3).






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