1. Why do we need to be able to describe mathematical systems?
Much of the mathematics we study is based on the properties described in this topic. For example when solving any equation, each step is carried out using on of these properties. Algebra, arithmetic, matrix and vector operations all use the basic properties of mathematical systems.
2. How can knowing about mathematical systems and properties help me to solve an equation?
Take the equation 2(x + 3) = 10
The first step involves using the distributive property:
2 × x + 2 × 3 = 10
2x + 6 = 10
The next step involves adding the inverse of 6 to both sides:
2x + 6 + -6 = 10 + -6
2x + 0 = 4 (0 is the identity element for addition)
2x = 4
The final step is to multipy both sides by the multiplicative inverse of 2 which is 1/2:
2x × 1/2 = 4 × 1/2
1 × x = 2 (1 is the identity element for multiplication)
x = 2
There are also a couple of steps using the associative property we could add too!
So although you may not realise, you are using quite a lot of mathematics when solving a simple equation!