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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: 57, 3, - 23 are the examples of rational numbers.
For example, each of the numbers 7, 35, 0.73, β25 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 β 0, is called an irrational number or incommensurable quantity.
Irrational numbers are the numbers which canβt be expressed in the form of p/q where p and q are integers and q β 0. Irrational numbers have infinite numbers of decimals of non-recurring nature.
Like: Ο, β2, β5 are the irrational numbers.
For example, each of the numbers β7, β3, 5β13 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 canβt be determined. Surds have infinite non-recurring decimals. Examples are β2, β5, β17 which are square roots or cube roots or nth root of any positive integer.
For example, each of the quantities β3, β7, β19, (16)^25 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 β9, β64, β(256/625) etc. expressed in the form of surds are commensurable quantities and are not surds (since β9 = 3, β64 = 4, β(256/625) = 45 etc.). In fact, any root of an algebraic expression is regarded as a surd.
Thus, each of βm, βn, 5βx2 etc. may be regarded as a surd when the value of m ( or n or x) is not given. Note that βm = 8 when m = 64; hence, in this case βm does not represent a surd. Thus, βm does not represent surd for all values of m.
β8 or β81 can be simplified into 2 or 3 which are rational numbers or positive integers, β8 or β81 are not surds. But the value of β2 is 1.41421356β¦., so the decimals continue up to infinite numbers and non-recurring in nature, so β2 is a surd. Ο 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 nβx is a surd of nth order when the value of nβx is irrational. In nβx 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 canβt 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 Ο and e, which are not the roots of algebraic expressions, are not surds.
Now we solve some problems on surds to understand more on surds.
1. Express β2 as a surd of order 4.
Solution
β2 = 212
=21Γ22Γ2
= 224
= 414
= 4β4
4β4 is a surd of order 4.
2. Find which are surds from the following numbers?
β24, β64 x β121, β50
Solution:
β24 = β4Γ6
= 2β2 Γ β3
So β24 is a surd.
β64 Γ β121 = 3β43 Γ β112
= 4 Γ 11
= 44
So β64 x β121 is rational and not a surd.
β50 = β2Γ25
= β2Γ52
= 5β2
So β50 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 12+β3 is expression, where the denominator is a surd.
12+β3
= 1Γ(2ββ3)(2+β3)Γ(2ββ3)
= (2ββ3)4β3
= 2 - β3
So the rationalizing factor of (2 + β3) is (2 - β3).
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