Why is the square root of 13 irrational?
It is irrational because you cannot express it as a ratio of two numbers. This is because it has a non-recurring, infinite decimal.
How do we add fractions that have different denominators?
When adding fractions, the denominators should be the same. So when we have different denominators, we have to do a little adjusting before we can add, and by this I mean we must find a common denominator. To do this, we use what is called a "least common multiple". For example, say we are adding 1/2 and 2/3. The least common multiple of these denominators is 6, because 6 is the lowest multiple of 2 and 3. We want the lowest multiple because it keeps the problem more simple. Now that we have found our least common multiple, we have to adjust the fractions to achieve a common denominator. To make 1/2 have a denominator of 6, what would we do? Multiply the fraction by 3. This would give us 3/6. Make sure when you do this step, that you do not change the value (e.g. 3/6 = 1/2). Next, we have to make 2/3 have the common denominator of 6. What would we do? Multiply the fraction by 2, giving us 4/6. So now our problem looks like this: 3/6 + 4/6. At this point, we are done adjusting the fractions and we can move on to adding them. When adding fractions, remember to only add the numerators, and keep the common denominator. So, to conclude, we would have 3/6 + 4/6 = 7/6.
When finding the slope between two points we use (y2 - y1) / (x2 - x1). Why does the order we write the numbers matter?
When subtracting, order matters because we are taking away one number from another number. So, if you do not pay attention to the order, you may change the value of the slope and thus end up with an incorrect answer. For example, let's look at the points (0,0) and (2,2). To find the slope, we use our (y2 - y1) / (x2 - x1). The calculation would go as follows: (2 - 0) / (2 - 0) = 2 / 2 = 1. The slope is 1. But if we do not pay attention to order, we may write something along the lines of (0 - 2) / (2 - 0). The calculation for this would go as follows: 0 / 2 = 0. The correct answer was 2, but changing the order gave us the incorrect answer of 0. That is why order matters when finding the slope between two points.