9th Class Mathematics Unit 1 Real Numbers Numericals with Solutions

1.1.1.Identify each of the following as a rational or irrational number: (i) 2.353535 (ii) 0.\overline{6} (iii) 2.236067… (iv) \sqrt{7} (v) e (vi) \pi (vii) 5+\sqrt{11} (viii) \sqrt{3}+\sqrt{13} (ix) 15/4 (x) (2-\sqrt{2})(2+\sqrt{2})

Result

(i) Rational

(ii) Rational

(iii) Irrational

(iv) Irrational

(v) Irrational

(vi) Irrational

(vii) Irrational

(viii) Irrational

(ix) Rational

(x) Rational

1.1.2.Represent the following numbers on a number line: (i) \sqrt{2} (ii) \sqrt{3} (iii) 4\tfrac{1}{3} (iv) -2\tfrac{1}{7} (v) \tfrac{5}{8} (vi) 2\tfrac{3}{4}

Geometric constructions (spiral of Theodorus for (i),(ii); subdivision of unit intervals for (iii)-(vi))

Number-line diagrams (no numeric result)

1.1.3.Express the following as a rational number p/q where p and q are integers and q \ne 0: (i) 0.\overline{4} (ii) 0.\overline{37} (iii) 0.\overline{21}

Given

(i) x = 0.overline{4} = 0.4444ldots

10x = 4.4444ldots ⇒ 10x - x = 4.4444ldots - 0.4444ldots

Result

9x = 4 ⇒ x = 49

Given

(ii) x = 0.overline{37} = 0.3737ldots

100x = 37.3737ldots ⇒ 100x - x = 37.3737ldots - 0.3737ldots

Result

99x = 37 ⇒ x = 3799

Given

(iii) x = 0.overline{21} = 0.2121ldots

100x = 21.2121ldots ⇒ 100x - x = 21.2121ldots - 0.2121ldots

Result

99x = 21 ⇒ x = 2199

1.1.4.Name the property used in the following: (i) (a+4)+b=a+(4+b) (ii) \sqrt{2}+\sqrt{3}=\sqrt{3}+\sqrt{2} (iii) x-x=0 (iv) a(b+c)=ab+ac (v) 16+0=16 (vi) 100\times1=100 (vii) 4\times(5\times8)=(4\times5)\times8 (viii) ab=ba

Result

(i) Associative property over addition

(ii) Commutative property over addition

(iii) Additive inverse

(iv) Left distributive property (of multiplication over addition)

(v) Additive identity

(vi) Multiplicative identity

(vii) Associative property under multiplication

(viii) Commutative property under multiplication

1.1.5.Name the property used in the following: (i) -3<-1 \Rightarrow 0<2 (ii) If a<b then \frac{1}{a}>\frac{1}{b} (iii) If a<b then a+c<b+c (iv) If ac<bc and c>0 then a<b (v) If ac<bc and c<0 then a>b (vi) Either a>b or a=b or a<b

Result

(i) Additive property

(ii) Reciprocal property

(iii) Additive property

(iv) Multiplicative property

(v) Multiplicative property

(vi) Trichotomy property

1.1.6.Insert two rational numbers between: (i) \frac{1}{3} and \frac{1}{4} (ii) 3 and 4 (iii) \frac{3}{5} and \frac{4}{5}

Formula

(i) q₁=12left(13+14right)=724,quad q₂=12left(724+14right)=1348

(ii) q₁=12(3+4)=72,quad q₂=12left(72+4right)=154

(iii) q₁=12left(35+45right)=710,quad q₂=12left(710+45right)=34

(i) 724,1348quad (ii) 72,154quad (iii) 710,34

Unit 1: Real Numbers — Exercise 1 1