O'k, the site has a focus on Algebra 1, and we're talking about fractions. Didn't we already cover this in 5th grade? Well - yes, and no. It turns out a lot of people still have problems with fractions. So, I thought it might be helpful to bring you up to speed. I'm calling this chapter -1 because, it's stuff you should have had before Algebra 1.
There's a couple of concepts you need to agree with before we get started. The first one is that if you cut a pie into 5 equal slices, and you still have all 5 of those slices, then you still have the hole pie. Mathematically that would be 5/5 = 1. In fact, we can expand that to say that anything divided by itself is equal to one.
Are you with me? O'k, lets try the second concept. If I have 1 of something, then I have that thing. Mathematically that would mean that multiplying any number times 1 doesn't change that number. 1*5 = 5.
If we can agree on these two facts - we're ready to start.
Fractions are also called ratios and both of these are just different ways to write division problems. 5 divided by 7 could be written as 5/7 or as 5:7. The 5 (on the top of the line) is called the numerator and the 7 (on the bottom of the line) is called the denominator.
Multiplication is really easy. You multiply the numerators together for the new numerator, and the denominators together for the new denominator. So, (5/7)(2/3) would be (5*2)/(7*3) = 10/21.
Note: When parenthesis touch - as in (5/7)(2/3) that means we're multiplying 5/7 times 2/3. Also, I'm using * as a multiplication symbol because we use x for other things in Algebra.
There's kind of a trick to division. If one fraction is divided by another, you flip the second and multiply it times the first. (5/7) /(2/3) would be (5/7)(3/2) or 15/14.
Now, if you like - that's all you need to know about division. But, if you want to know why that works - keep reading.
What we're actually doing is multiplying the inverse of the denominator times the numerator and denominator. So, in our example we have (5/7)/(2/3) and we're going to multiply that times (3/2)/(3/2). Hopefully you noticed that (3/2)/(3/2) is a number divided by itself. So, it equals 1, and 1 times anything is just that thing. So we can multiply by this and not change the value.
So, if we're multiplying these two together, we multiply the numerator times the numerator and get (5/7)(3/2). And we multiply the denominators and get (2/3)(3/2) which is (2*3)/(3*2) which is 6/6 which is 1. If we divide a number by 1, we just get the original number. So, the denominator drops to 1. The numerator (5/7)(3/2) becomes our original problem with an answer of 15/14.
Subtraction is just addition with negative numbers. Addition with fractions is kind of a pain.
You need to have the same denominators for each of the ratios. So, if I'm adding 1/3 to 1/4, I have to change the denominators to something that is evenly divisible by both 3 and 4. In this case, I'd go with 12. Now, remember the rule where we can multiply anything by 1 and not change it's value? And the rule that anything divided by itself is 1. So, we're going to multiply (1/3) by (4/4) and get 4/12. Can we agree that 4/12 is the same as 1/3? O'k, then we multiply 1/4 times 3/3 we get 3/12 which is the same as 1/4.
Now we can add 4/12 + 3/12 = 7/12.
XVII.. Square Root Functions
XVIII. Absolute Value Functions