Explain why the properties of real numbers are important to know when working with algebra?
In beginning Algebra, we let the variables stand for some real number. Thus the properties of real numbers can be used when dealing with the variables.
Some important properties:
commutative property: `2*3=3*2` so `(x+2)x=x(x+2)`
associative property: (2+3)+5=2+(3+5) so (x+3)+5=x+(3+5)
identity: 3+0=0+3=3 x+0=0+x=x This idea is used when studying quadratics and learning to complete the square.
`3*1=1*3=3` and `x*1=1*x=x` I learned to call certain things FFOO's (pronounced fahfooh) or a fancy form of 1. You use this idea wn rationalizing, finding common denominators, and many other places.
inverse: 3+(-3)=0 and x+(-x)=0
`3*1/3=1` and `x*1/x=1,x!=0` We use these all of the time to transform equations. We add/subtract the same thing to both sides of an equation (or multiply/divide). We are then left with an addition of zero or multiplication by 1. (This is not frequently emphasized.)
e.g. x+5=7 Subtract 5 (or add -5) to both sides to get x+5-5=7-5 or x+0=2 (using the inverse property) or x=2 (using the identity property.)
distributive property: 2(3+5)=2(3)+2(5) so 2(x+3)=2x+6 used on virtually every problem.
These properties form the logical basis that allows you to perform operations on the numbers and variables you encounter.