Properties Worksheets Properties Worksheets In mathematics we use many types of numbers such as natural, whole, integer, positive, negative and many more. Real numbers are the collection of all numbers, rational and irrational numbers are also comes under in it. All the properties of numbers worksheets are classified according to operations. We can say that the properties are in the respect of real numbers. Let’s take a close look on some properties… Properties of addition on R 1. Closure property: - The sum of two real numbers is always a real number. That is a ?R, b?R => a + b ?R for all a, b ?R. 2.
Associative law: - (a + b) + c = a + (b + c) for all a, b, c ?R.
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3.
Commutative law: - a + b = b + a for all a, b ?R.
4. Additive identity: - Zero is a real number such that 0 + a = a + 0 = a for all a ?R. o is called the additive identity in R. 5.
Additive inverse: -
For each a ?R, there exist negation a ?R such that a + (-a) = (-a)
Properties of multiplication on R 1.
R is closed for multiplication. That is a ?R, b?R => ab ?R for all a, b ?R.
2.
(ab) c = a (bc) for all a, b, c ?R.
3.
Ab = ba for all a, b ?R.
4.
A (b+ c) = ab + bc
5.
I ?R such that I.a = a.I = a for all a ?R. I is called the multiplicative identity in R.
[Associative law] [Commutative law] [Distributive law]
6. For each non- zero real number a ?R there exist real number 1 / a such that a .1 /a = 1 / a = 1. The number 1 / a is called the multiplicative inverse or reciprocal of a. Properties of subtraction and division on R 1.
R is closed for subtraction.
2.
Subtraction on R does not satisfy the commutative and associative laws.
3.
R is not closed for division, since 2 ?R, 0 ?R but 2 / 0 ! ?R.
An Important Property: - Between any two rational numbers, there are an infinite number of rational numbers.
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If you talk about the properties of irrational numbers, then we’ll get: - The set P of all irrationals is not closed for addition, since the sum of two irrationals need not to be irrational. The set P of all irrationals is not closed for multiplication, since the products of two irrationals need not to be irrational. Some more properties of real numbers: 1. Factors and multiples: - for real numbers a & b, we say that a is the factor of b, if b = ac for some real number c and we write, a / b. If a / b, then b is called a multiple of a. 2. Ordering: - For real number a & b, we say that a is greater than b, if a = b + c for some c ? R, and we write a > b. Although we have so many properties but all properties cannot be discuss here.
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