Conjugate Surds

In this section, we will discuss about conjugate surds. For the basics of surds, please visit the page an introduction to surds.

Definition of Conjugate Surds

Mathematically, if x=a+√b where a and b are rational numbers but √b is an irrational number, then a-√b is called the conjugate of x.

Thus we can define conjugate surds as follows:

A surd is said to be a conjugate surd to another surd if they are the sum and difference of two simple quadratic surds. In other words, the sum and the difference of two simple quadratic surds are conjugate to each other.

For example, we consider two simple quadratic surds $\sqrt{2}$ and $7\sqrt{3}.$ According to the above definition, the two binomial surds $\sqrt{2}+7\sqrt{3}$ and $\sqrt{2}-7\sqrt{3}$ are conjugate (or complementary) to each other. In a similar way, we have the following examples of conjugate surds:

(i) $5\sqrt{2}+2\sqrt{7}$ and $5\sqrt{2}-2\sqrt{7}$

(ii) $1+\sqrt{3}$ and $1-\sqrt{3}$

Note: In general, surds of the forms (a – y√b) and (a + y√b) are complementary/conjugate to each other.

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Properties of Conjugate Surds

• The general form of two conjugate surds are $a\sqrt{x}+b\sqrt{y}$ and $a\sqrt{x}-b\sqrt{y}.$

• The product of two conjugate surds is always a rational number.

Proof. The product of two general conjugate surds is given by

$(a\sqrt{x}+b\sqrt{y})$$(a\sqrt{x}-b\sqrt{y})$

$=(a\sqrt{x})^2-(b\sqrt{y})^2$ $[\because (m+n)(m-n)=m^2-n^2]$

$=a^2x-b^2y,$

which is a rational number.

The sum of two conjugate surds is always a rational number.

Proof. 

As conjugate surds have the form a+√b and a-√b where both a and b are rational numbers, the sum will be equal to

(a+√b)+(a-√b)

= a+√b+a-√b

= 2a, which is a rational number.

The difference of two conjugate surds is not a rational number.

Proof. 

The difference of two conjugate surds a+√b and a-√b where both a and b are rational numbers is

(a+√b)-(a-√b)

= a+√b-a+√b

= 2√b, which is an irrational number.

 

Importance of Conjugate Surds

To rationalize the denominator of a fraction containing surds, we need to take the help of conjugate surds. In this case, we need to multiply the denominator with its conjugate surd. For example,

Question: Rationalize the denominator of $\dfrac{1}{1+\sqrt{2}}$

Answer:

See that the denominator 1+√2 is not a rational number. To rationalize the denominator we have to multiply with the conjugate of 1+√2 which is 1-√2. By doing so we get that

$\dfrac{1}{1+\sqrt{2}}=\dfrac{1-\sqrt{2}}{(1+\sqrt{2})(1-\sqrt{2})}$

$=\dfrac{1-\sqrt{2}}{1^2-\sqrt{2}^2}$ $[\because (a-b)(a+b)=a^2-b^2]$

$=\dfrac{1-\sqrt{2}}{1-2}$

$=-(1-\sqrt{2})=-1+\sqrt{2}$

 

Note: Conjugate of surds are also very useful in many branches of Mathematics; like Calculus, Geometry, etc.

 

Question Answer on Conjugate Surds

Question 1: Find the conjugate surd of √5-√2.

Solution:

See that the surd √5-√2 is the difference of the two surds √5 and √2. So the desired conjugate surd will be the sum of the two surds. Hence, the conjugate of √5-√2 is √5+√2.

 

Question 2: Find the conjugate surd of √3 -2.

Solution:

Note that the surd √3 -2 can be written as -2+√3. As we know that the conjugate surd of a+√b (a rational and √b irrational) is a-√b, so the conjugate of -2+√3 will be -2-√3.

 

Question 3: Find the conjugate surd of √5-2.

Solution:

We have √5-2 = -2+√5.

The conjugate of a-√b is a+√b.   

Thus, the conjugate of -2+√5 is -2-√5.

 

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FAQs on Conjugate Surds

Q1: What are conjugate surds?

Answer: A pair of surds is called conjugate of each other if their forms are x√a-y√b and x√a+y√b where a, b, x, and y are rational numbers and both a, b are square-free.

Q2: Give an example of conjugate surds.

Answer: The conjugate surd of √2 is -√2; and vice-versa.