Operations with Real Numbers

Adding Real Numbers

Properties

1. Closure:

The result of adding two real numbers is another real number.

A + b PerteneceReal Number

pi + Golden Section PerteneceReal Number

2. Associative :

The way in which the summands are grouped does not change the result.

(a + b) + c = a + (b + c)

Real Number Properties

3. Commutative :

The order of the addends does not change the sum.

a + b = b + a

Real Number Properties

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

pi + 0 = pi

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.

−(−Golden Section) = Golden Section

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.

Rule of Signs

Properties

1. Closure:

The result of multiplying two real numbers is another real number.

a · b PerteneceReal Number

2. Associative:

The way in which the factors are grouped does not change the result.

(a · b) · c = a · (b · c)

(e · pi ) · Golden Section = e · (pi · Golden Section)

3. Commutative:

The order of factors does not change the product.

a · b = b · a

Real Number Properties

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

pi · 1 = pi

5. Multiplicative inverse:

A number is the reciprocal of another if when multiplied by each other, the product is the muliplicative identity.

Real Number Properties

Real Number Properties

6. Distributive:

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

pi · (e + Golden Section) = pi · e + pi · Golden Section

Removing a common factor:

It is the reverse of the distributive property.

a · b + a · c = a · (b + c)

pi · e +pi · Golden Section = pi · (e + Golden Section)

Dividing Real Numbers

The division of two real numbers is defined as the product of the dividend by the reciprocal of the divisor.