Notes-Class-9-Mathematics-1-Chapter-2-Real Numbers-Maharashtra Board

Number System :

(i) Natural numbers : The counting numbers 1, 2, 3, 4, ... are called natural numbers.

• These numbers are written in the set form as N = {1, 2, 3, 4, ... }
• The set of natural numbers is adequate for addition and multiplication but not for subtraction and division.
• Examples : 2 + 5 = 7, 2 × 5 = 10. Here, 7 and 10 are natural numbers but 2 −5 ∉ N; 2 ÷ 5 ∉

(ii) Whole numbers : The union of set of natural numbers and zero is a set of whole numbers. It is denoted by W.

• These numbers are written in the set form as W = {0, 1, 2, 3, ... }

(iii) Integers : The set of natural numbers, zero and opposite of all natural numbers is called set of integers and denoted by I.

• These numbers are written in the set form as I = { ... , −3, −2, − 1, 0, 1, 2, 3, ... }
• The order relation on I on the number line is defined as follows : ... , −3 < − 2 < − 1 < 0 < 1 < 2 < 3, ...
• The set of integers is adequate for addition, subtraction and multiplication but not for division. e.g. 3 ÷ 9 ∉

(iv) Rational Numbers : If p and q are integers (q ≠ 0), then the number \(\frac{p}{q}\) is called a rational number

• The set of rational numbers is denoted by Q.
• Q = { \(\frac{p}{q}\) | p, q ∉ I and q ≠ 0}      ……(Rule method)
• The set of real numbers is denoted by R.

Know This : (i) In the definition of rational numbers, the condition that q ≠ 0 is necessary, since division by zero is not defined. (ii) Integers are also rational numbers because every integer p = \(\frac{p}{1}\). Hence, I ⊆ Q. (iii) N ⊆ W ⊆ I ⊆ Q ⊆ R.

 Properties of rational numbers :

(A) Properties of addition of rational numbers :

If a, b, c are any rational numbers, then

(i) Commutative property : a + b = b + a

(ii) Associative property : (a + b) + c = a + (b + c)

(iii) Additive identity : a + 0 = 0 + a = a

• 0 is called the identity element for addition]

(iv) Additive inverse : a + ( — a) = 0

• For two rational numbers to be additive inverses of each other: their sum should be 0.

(B) Properties of multiplication of rational numbers :

(i) Commutative property : a × b = b × a  ()

(ii) Associative property : (a × b) × c = a × (b × c)

(iii) Multiplicative identity : a × 1 = 1 × a = a

• 1 is called the identity element for multiplication.

(iv) Multiplicative inverse : a × \(\frac{1}{1}\)  = \(\frac{1}{a}\) × a = 1 (a ≠ 0)

• For two rational numbers to be multiplicative inverses of each other; their product should be 1.

 

Know This : Decimal Representation of Rational Numbers : Any positive rational number \(\frac{p}{q}\) after actual division can be expressed as \(\frac{p}{q}=m+\frac{r}{q}\) , where m is a non−negative integer and 0 ≤ r < q. In the decimal system, m is the integral part and (\frac{p}{q}\) is the fractional part. The decimal representation of rational numbers is either (i) terminating or (ii) non−terminating recurring. Example : If the numerator 3 of the rational number \(\frac{3}{5}\) is divided by 5, it yields terminating decimal 0.6. If the denominator of a rational number contain the prime numbers 2 or 5 or 2 and 5, the process of division terminates after a finite number of steps. The decimal form of \(\frac{2}{3}\) is 0.6666... . This is non−terminating recurring form. Here, digit 6 is recurring. Its decimal representation is 0. . Similarly, the decimal form of \(\frac{3}{7}\) is 0428571428571… . It is written as 0.

To express the recurring decimal in \(\frac{p}{q}\)  form

Ex. (1) Express the recurring decimal 0.777.... in \(\frac{p}{q}\)  form.

Ex. (2) Express the recurring decimal 7.529529529.... in \(\frac{p}{q}\)  form.

Remember : (1) Note the number of recurring digits after decimal point in the given rational number. Accordingly multiply it by 10, 100, 1000 e.g. In \(2.\dot 3\), digit 3 is the only recurring digit after decimal point, hence to convert \(2.\dot 3\) in \(\frac{p}{q}\) form multiply \(2.\dot 3\) by 10. In \(1.\overline{24}\)  digits 2 and 4 both are recurring therefore multiply \(1.\overline{24}\)   by 100. In \(1.\overline{513}\) digits 5, 1 and 3 are recurring so multiply \(1.\overline{513}\) by1000. (2) If the prime factors of the denominator of a rational number are 2 or 5 only then its decimal expansion is terminating. If the prime factors are other than 2 or 5 also then its decimal expansion is non terminating and recurring.

Irrational and real numbers

Irrational number :

• The number which is not rational is called an irrational number.
• The set of irrational numbers is denoted by 'Q’.
• The decimal representation of an irrational number is non−terminating, non−recurring.

Real Numbers :

• The numbers which are represented by points on a number line are real numbers.
• Every point on the number line is associated with a unique real number and every real number is associated with a unique point on the number line.
• Every rational number is a real number. But every real number may not be a rational number.
• Numbers \(\sqrt{2},\sqrt{3},\sqrt{5}\) .. etc. can be shown on a number line but these are not the rational numbers.

Proof : \(\sqrt{2}\)  is irrational number.

This can be proved using indirect proof.

Let us assume that \(\sqrt{2}\) is rational. So \(\sqrt{2}\) can be expressed in \(\frac{p}{q}\) form.

\(\frac{p}{q}\)  is the reduced form of rational number hence p and q have no common factor other than 1.

\(\sqrt{2}\) = \(\frac{p}{q}\),  ∴  2 = \(\frac{p^2}{q^2}\)    …..(Squaring both the sides)

∴  2q² = p²

∴  p² is even.

∴  p is also even means 2 is a factor of p. ....(I)

∴  p = 2t ∴ p² = 4t²     …t ∈ I

∴  2q² = 4t² ( ∴ p² = 2q²) ∴ q² = 2t² ∴ q2 is even. ∴ q is even.

∴  2 is a factor of q. .... (II)

From the statement (I) and (II), 2 is a common factor of p and q both.

This is contradictory because in \(\frac{p}{q}\) ; we have assumed that p and q have no common factor except 1.

∴ Our assumption that \(\sqrt{2}\) is rational is wrong.

∴ \(\sqrt{2}\) is irrational number.

Similarly, one can prove that \(\sqrt{3}\), \(\sqrt{5}\)  are irrational numbers.

Numbers \(\sqrt{2},\sqrt{3},\sqrt{5}\) can be shown on a number line.

We know that every rational number is a real number. But \(\sqrt{2},\sqrt{3},-\sqrt{5}\) π, 3 +\(\sqrt{5}\) etc. are not rational numbers. It means that Every real number may not be a rational number.

Decimal form of irrational numbers :

Let us find the square root of 2 using division method :

∴ \(\sqrt{2}\) = 1.41421…

In the above division, numbers after decimal point are unending, means it is non−terminating. Even no group of numbers or a single number is repeating in its quotient. So decimal expansion of such numbers is non terminating, non−recurring.

• A number whose decimal expansion is non−terminating, non−recurring is irrational.

Properties of irrational numbers

• Addition or subtraction of a rational number with irrational number is an irrational number.
• Multiplication or division of non zero rational number with irrational number is also an irrational number.
• Addition, subtraction, multiplication and division of two irrational numbers can be either a rational or irrational number.

Properties of order relation on Real numbers :

• If a and b are two real numbers then only one of the relations holds good.
• e. a = b or a < b or a > b
• If a < b and b < c then a < c
• If a < b then a + c < b + c
• If a < b and c > 0 then ac < bc and If c < 0 then ac > bc

Square root of a Negative number :

We know that, if \(\sqrt{a}\) = b then b² = a.

Hence if \(\sqrt{5}\) = x then x² = 5.

Similarly we know that square of any real number is always non−negative. It means that square of any real number is never negative.

But ( \(\sqrt{-5}\) )² = − 5, ∴ \(\sqrt{-5}\) is not a real number.

Hence square root of a negative real number is not a real number.

Root of positive rational number :

If n is a positive integer and xⁿ = a, then x is the n^th root of a . x = \(\sqrt [n] {a}\) . This root may be rational or irrational.

For example, 2⁵ = 32, ∴ 2 is the 5th root of 32, but if x⁵ = 2 then x = \(\sqrt [5] {2}\) , which is an irrational number.

Surds :

If n is an integer greater than 1 and if n^th root of a is x, and x is a positive real number, then it is written as xⁿ = a  or \(\sqrt [n] {a}\) = x.

If a is a positive rational number and n^th root of a is x and if x is an irrational number, then x is called a surd.

In the surd \(\sqrt [n] {a}\) the symbol √¯ is called radical sign, n is called the order of the surd and a is called the radicand.

Examples :

(1) If a = 7, n = 3, then \(\sqrt [3] {7}\) is a surd because \(\sqrt [3] {7}\) is an irrational number.

(2) If a = 27, n = 3 then \(\sqrt [3] {27}\) is not a surd because \(\sqrt [3] {27}\) = 3 is not an irrational number.

(3) \(\sqrt [3] {8}\) is a surd or not ?

Let  = p, ∴ p³ = 8. We know 2 is cube−root of 8.

2 is not an irrational number. .

∴ \(\sqrt [3] {8}\)  is not a surd.

• This year we are going to study surds of order 2 only, means
• The surds of order 2 is called Quadratic surd.

Simplest form of a Surd :

A surd of order 'n' is said to be in its simplest form if it has the following properties:

(1) The radicand has no factor which is the nth power of a rational number.

(2) The radicand is not a fraction.

(3) 'n' is the least order of the surd.

e.g. (i) 27 = \(\sqrt {9×3}\) = \(\sqrt {9}×\sqrt {9}\) = \(3\sqrt {3}\)

(ii) 50 = \(\sqrt {25×2}\) = \(\sqrt {25}×\sqrt {2}\) = \(5\sqrt {2}\)

\(\sqrt {2}\),\(\sqrt {3}\), \(\sqrt {5}\) .... these type of surds are in the simplest form which cannot be simplified further.

Similar or like surds :

If p and q are rational numbers then p\(\sqrt {a}\), q\(\sqrt {a}\)  are called like surds. Two surds are said to be like surds if their order is equal and radicands are equal.

e.g. 3\(\sqrt {2}\), −4\(\sqrt {2}\), \(\frac{5}{3}\sqrt {2}\)   are some like surds.

[Note : 3\(\sqrt {2}\), −4\(\sqrt {5}\), \(\frac{5}{3}\sqrt {3}\)   are not some like surds.]

Question : \(\sqrt {45}\) and \(\sqrt {80}\) are the surds of order 2, so their order is equal but radicands are not same, Are these like surds?

Remember : In the simplest form of the surds if order of the surds and redicand are equal then the surds are similar or like surds.

 Comparison of surds :

Two surds of the same order can be compared by comparing the radicands.

Let a and b be two positive real numbers.

If a < b, then a × a < b × a.

∴ a² < ab      ….(1)

If a < b, then a × b < b × b.

∴ ab <  b²    ….(2)

From (1) and (2), a² < ab < b².

∴ a² < b²

If a > b, then a² > b² and if

a = b, then a² = b².

Question : Compare the surds \(2\sqrt {7}\) and \(5\sqrt {3}\) .

Operations on like surds :

Mathematical operations like addition, subtraction, multiplication and division can be done on like surds.

Examples :

(i) 7\(\sqrt {3}\) + 29\(\sqrt {3}\)  = (7 + 29)\(\sqrt {3}\)  = 36\(\sqrt {3}\)

(ii) 7\(\sqrt {5}\) − 29\(\sqrt {5}\)  = (7 − 29)\(\sqrt {5}\)  = −22\(\sqrt {5}\)

(iii) 7\(\sqrt {5}\)  × 2\(\sqrt {7}\)  = (7 × 2)\(\sqrt {5×7}\)  = 14\(\sqrt {35}\)

(iv) 21\(\sqrt {125}\)  ÷ 3\(\sqrt {5}\)  = \(\frac{21\sqrt {125}}{3\sqrt{5}}\)

= \(7\sqrt{\frac{125}{5}}\) = \(7\sqrt {25}\)  = 7 × 5 = 35  ..(rational number)

Note that product and quotient of two surds may be rational numbers.

Rationalization of surds :

If the product of two surds is a rational number, then each of the two surds is called a rationalizing factor of the other.

e.g. the product of the surds \(\sqrt {2}\) and \(\sqrt {8}\) is \(\sqrt {16}\) = 4, which is a rational number.

\(\sqrt {2}\) is the rationalizing factor of \(\sqrt {8}\) and \(\sqrt {8}\) is the rationalizing factor of \(\sqrt {2}\).

Question : Find the rationalizing factor of \(\sqrt {27}\)

Question : Rationalize the denominator of \(\frac{1}{\sqrt {5}}\)

Remember : The rationalizing factor of a given surd is not unique. If a particular surd is a rationalizing factor of a given surd, then the product of this factor and a rational number is also the rationalizing factor of the given surd. It is convenient to use the simplest of all rationalizing factors of a given surd. It is easy to use the numbers with rational denominators. That is why we rationalize it.

 Binomial quadratic surd :

The sum or difference of two numbers one of which is a quadratic surd, the other either a non−zero rational number or a quadratic surd is called binomial quadratic surd.

Examples :

(1) (i) a + x\(\sqrt {b}\)  ; (ii) a − x\(\sqrt {b}\)  ; (iii) x\(\sqrt {a}\) + y\(\sqrt {b}\)  (iv) x\(\sqrt {a}\) − y\(\sqrt {b}\) are the binomial expressions of quadratic surds,

where \(\sqrt {a}\), \(\sqrt {b}\) are quadratic surds. a, b, x and y are non−zero rational numbers.

(2) (i) 5 + 3\(\sqrt{2}\)  (ii) 3 − 2\(\sqrt {3}\)  (iii) 2\(\sqrt {3}\)  + 3\(\sqrt {5}\)  (iv) 3\(\sqrt {2}\)  — 5\(\sqrt {3}\)  are also the examples of binomial quadratic surds.

The product of two binomial surds \((\sqrt {3}-\sqrt {2})\) and \((\sqrt {3}+\sqrt {2})\) is \((\sqrt {3})^2-(\sqrt {2})^2\) = 3 – 2 = 1.

1 is a rational number. Hence  \((\sqrt {3}-\sqrt {2})\) and \((\sqrt {3}+\sqrt {2})\)  are the conjugate pairs of each other.

The product of two conjugate binomial surds is a rational number.

Rationalization of the denominator :

The product of two conjugate binomial surds is always a rational number. Using this property, the rationalization of the denominator in the form of binomial surd can be done.

Example :

(1) Rationalize the denominator : \(\frac{5}{3\sqrt{2}+\sqrt{5}}\)

The conjugate pair of \(3\sqrt{2}+\sqrt{5}\) is \(3\sqrt{2}-\sqrt{5}\)

\(\frac{5}{3\sqrt{2}+\sqrt{5}}\) = \(\frac{5}{3\sqrt{2}+\sqrt{5}}\) × \(\frac{3\sqrt{2}-\sqrt{5}}{3\sqrt{2}-\sqrt{5}}\)

= \(\frac{5(3\sqrt{2}-\sqrt{5})}{(3\sqrt{2})^2-(\sqrt{5})^2}\)

= \(\frac{15\sqrt{2}-5\sqrt{5})}{9×2-5}\)

= \(\frac{15\sqrt{2}-5\sqrt{5})}{13}\)

Absolute value :

If x is a real number then absolute value of x is its distance from zero on the number line which is written as | x |, and | x | is read as Absolute Value of x or modulus of x.

• If x > 0 then | x | = x If x is positive then absolute value of x is
• If x = 0 then | x |= 0 If x is zero then absolute value of x is zero.
• If x < 0 then | x | = −x If x is negative then its absolute value is opposite of

Examples:  (1) |3| = 3, |−3| = −(−3) = 3, |0| = 0

The absolute value of any real number is never negative.