subject: Real Numbers [print this page] A combined set of rational numbers and irrational numbers, is called the set of Real numbers.
What is a real number?
By definition, a number that can be plotted on a number line is called a real number. A real number is either a rational or an irrational number. Thus, the real number 2 is a rational number, while the real number 2 is an irrational number. A real number can be thought of a point on an infinitely long number line, where in the integers are equally spaced. And any point on that line would represent a real number.
1. Closure: Real numbers are closed under addition and multiplication. That means a+b = real number and a*b = real number.
2. Commutative: For real numbers a and b, a+b = b+a and a*b = b*a.
3. Associative: For real numbers a and b , (a+b)+c = a+(b+c) and (a*b)*c = a*(b*c)
4. Identity: In addition, a+0 = 0+a = a i.e., 0 is the identity element for the operation of addition. In multiplication, 1*a = a*1 = a i.e., 1 is the identity element for the operation of multiplication.
5. Inverse: For each a, there is a unique real number (-a), called the additive inverse of a such that a+(-a) = (-a)+a = 0. For each a " 0, there is a real number 1/a, called the multiplicative inverse of a such that a * (1/a) = (1/a) *a = 1.
6. Distributive: a*(b+c) = ab + ac and (b+c)*a = ba + ca
7. Order: The real number system is ordered, i.e., if a and b are different real numbers, then either a b.
8. Density: The real number system is dense, that is, between any two distinct real numbers there are infinitely many more real numbers.
9. Completeness: The real number system is complete.
Adding real numbers or subtracting real numbers:
When adding or subtracting real numbers, the rational parts are added and the like irrational parts are combined. Everything else is left as it is. For example: 1+2+3"(2) +"5 = 1 + 4"(2) +"5
Example: Width of a field is 2 less than its length. If length = "10 then find the perimeter.
