Operations with Real Numbers
Adding Real Numbers
The result of adding two real numbers is another real number.
A + b
2. Associative :
The way in which the summands are grouped does not change the result.
(a + b) + c = + (b + c)
3. Commutative :
The order of the addends does not change the sum.
a + b = b + a
4. Additive identity:
The 0 is the neutral element in the addition because every number added to it gives the same number.
a + 0 = a
+ 0 =
5. Additive inverse:
Two numbers are opposites if they are added together and the result is zero.
a + (−a)= 0
e − e = 0
The opposite of the opposite of a number is equal to the same number.
Subtracting Real Numbers
The difference of two real numbers is defined as the sum of the minuend plus the opposite of the subtrahend.
a − b = a + (−b)
Multipying Real Numbers
The rule of signs for the product of integers and rational numbers is still maintained with the real numbers.
The result of multiplying two real numbers is another real number.
a · b
The way in which the factors are grouped does not change the result.
(a · b) · c = a · (b · c)
(e · ) · = e · ( · )
The order of factors does not change the product.
a · b = b · a
4. Multiplicative Identity:
The 1 is the neutral element of the multiplication because any number multiplied by it gives the same number.
a · 1 = a
· 1 =
5. Multiplicative inverse:
A number is the reciprocal of another if when multiplied by each other, the product is the muliplicative identity.
The product of a number for a sum is equal to the sum of the products of this number for each of the addends.
a · (b + c) = a · b + a · c
· (e + ) = · e + ·
Removing a common factor:
It is the reverse of the distributive property.
a · b + a · c = a · (b + c)
· e + · = · (e + )
Dividing Real Numbers
The division of two real numbers is defined as the product of the dividend by the reciprocal of the divisor.