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Number system
The document discusses different types of number systems including real numbers, rational numbers, irrational numbers, integers, fractions, natural numbers, even numbers, and odd numbers. It defines these number systems and provides examples. It also outlines several key properties of real numbers including closure, identity, inverse, commutativity, associativity, and distributivity with regards to addition and multiplication. Finally, it presents several exercises involving identifying properties, finding inverses, and determining which number sets have certain properties like closure, identity, and inverse.
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Number system
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- 2. Mathematics. Chapter: Number Systems. NumberSystem. Number system is the way to represent a number in different forms. Real Numbers. The sum of rationaland irrationalnumbers is calledreal numbers. Real number = RationalNumbers+ IrrationalNumbers. Denoted By Capital R Rational Numbers. A rationalnumber is a number which can be put in the form 𝑃/q) where p, q ϵ Z ∧ q ≠ 0. The number√16, 3, 7, 4 etc. are rational numbers. √16 Can be reduced to form 𝑝 𝑞⁄ where p, q ϵ Z and q ≠ 0 because √16 =4=4 1⁄ . Denoted by Capital Q.
- 3. IrrationalNumbers. Irrational Numbers arethose numbers which cannot be put into the form 𝑝 𝑞⁄ where p, q ϵ Z and q≠0. The numbers√2, √3, √5, √5 16 . Denoted By Capital 𝑄, (Q Dash) Fractions. Numbers that can be written as ratios of 2 integers E.g. 2 3 , 6 8 , 2 7 , 3 5 . Integers. Positive Integers + 0 (Zero) + Negative Integers. Denoted By CapitalZ. Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …} Negative Integers. Denoted by 𝑍′ (Z Dash) 𝑍′ = {-1,-2, -3, -4, -5, -6, …} Natural Numbers. N ={1, 2, 3, 4, 5, 6, 7, …} Whole Numbers.
- 4. W = {0,1, 2, 3, 4, 5, 6, 7,8, …} Even Numbers. E= {0, 2, 4, 6, 8, 10, 12, 14, …} Odd Number. O ={1, 3, 5, 7, 9, 11, 13, …} Properties of Real Numbers. Addition Laws: 1. Closure Property w.r.t addition. ∀ a, b ϵ R,a+b=R (∀ Stands for “for all”) 2. Associative propertyw.r.t addition. ∀ a, b, c ϵ R, a+(b+c)=(a+b)+c 3. Additive identity. ∀ a ϵ R , ∃ 0 ϵ R such that a+0=0+a=a (∃ stand for “there exist”) 0 (read as zero) is calledthe identity element of addition. 4. Additive inverse. ∀ a ϵ R, ∃ (-a) ϵ R such that a+(-a)=(-a)+a=0 5. Commutative Property w.r.t addition. ∀ a, b ϵ R, a+b=b+a
- 5. Multiplicative Laws: i. ClosureProperty w.r.t Multiplication. ∀ a, b ϵ R,a.b=R (a.b is usually written as ab) ii. Associative propertyw.r.t multiplication. ∀ a, b, c ϵ R, a(bc)=(ab)c iii. Multiplicative identity. ∀ a ϵ R , ∃ 1 ϵ R such that a.1=1.a=a 1 is called the multiplicativeidentity of real numbers. iv. Multiplicative inverse. ∀ a(≠0) ϵ R, ∃ (𝑎−1 ) ϵ R such that a. 𝑎−1 = 𝑎−1 .a=1 (𝑎−1 is also written as 1 𝑎 ) v. Commutative Property w.r.t addition. ∀ a, b ϵ R, ab=ba DistributivePropertyof Multiplicationoveraddition. ∀ a, b, c ϵR such that; a (b+c)= ab+ac ( left D.Law) (a+b) c= ac+bc ( Right D.Law)
- 6. Exercise. Q No 1. Nameof the properties of real numbers given below. a. 25+48=48+25 b. √11 ×√3=√3×√11 c. √5(3+7)=√5 3 +√5 7 d. ( 11 2 + 9 5 ) + 7 10 = 11 2 +( 9 5 + 7 10 ) e. 0 + 4 5 = 4 5 + 0 f. ( √5 2 × 1) = √5 2 Q No 2. Write the additive and multiplicativeinverse of the following. a. 1 𝑎 b. 1 c. √57 d. 0 e. √3 7 f. √5 − 2 g. −7
- 7. Q No 3. Whichof the followingset have the closure property w.r.t addition and multiplication. a. 0 b. {0,-1} c. {0, 1} d. {-1, -2, -3, …} e. Set of real numbers. f. Set of rational numbers. g. Set of whole numbers. Q No 4. Which of the followingset have additive and multiplicative identity. a. Set of real numbers. b. Set of whole numbers. Q No 5. Which of the followingsets have the additive and multiplicativeinverse. a. Set or rational numbers. b. Set of prime numbers. c. Set of real numbers. d. Set of integer numbers.