Real number whose square is non-negative is called real number. A number whose square is negative is said to be
imaginary number thus ⃭√- 1, √- 2, √-3, etc., are imaginary numbers, since the square of each one of them is negative.
In fact, all rational and all irrational numbers form the set of all real numbers.
Remarks: Every real number is either rational or irrational. Consider any real number.
(i) If it has a terminating or repeating decimal representation, or if it is an integer, then it is rational.
(ii) If it has a non-terminating, non-repeating decimal representation, then it is irrational.
Additional properties of Real numbers
(i) Closure Property: The sum of two real numbers is always a real number.
(ii) Associative Law: (a + b) + c = a + (b + c) for all real numbers a, b and c.
(iii) Commutative law a + b = b + a for all real numbers a and b.
(iv) Existence of Additive Identity: Clearly, 0 is a real number such that
0 + a = a + 0 = a for every real numbers.
So 0 is called the additive identity for real number.
(v) Existence of Additive Inverse: For each real number a, there exists a real number (- a) such that a + (-a) = 0
a and (-a) are called the additive inverse (or negative) of each other.
(iii) Commutative law a + b = b + a for all real numbers a and b.
(iv) Existence of Additive Identity: Clearly, 0 is a real number such that
0 + a = a + 0 = a for every real numbers.
So 0 is called the additive identity for real number.
(v) Existence of Additive Inverse: For each real number a, there exists a real number (- a) such that a + (-a) = 0
a and (-a) are called the additive inverse (or negative) of each other.
Multiplication properties of real numbers:
(i) Closure property: The product of two real numbers is always a real number.
(ii) Associate law: (ab)c = a(bc) for all real numbers a, b and c.
(iii) Commutative law ab = ba for all real numbers a and b.
(iv) Existence of additive identity: Clearly, 1 is a real number such that 1.a = a.1 = a for every real number a.
1 is called the multiplicative identity for real numbers.
(v) Existence of Multiplicative inverse: For each nonzero real number
a, there exists a real number (1/a) such that a.1/a = 1/a .a =1.
a and 1/a are called the multiplicative inverse (or reciprocal) of each other.
(vi) Distributive Law Of Multiplication Over Addition: We have
a(b + c) = ab + ac and (a +b)c = ac + bc for all real numbers a, b and c.
Completeness of property: On the number line, each point corresponds to a unique real number. And , every real number can be represented by a unique point on the number line.
Density property: Between any two real numbers there exist infinitely many numbers.
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