Adding & subtracting rational numbers
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- We're told to look at the rational numbers below, order them from least to greatest. They really didn't have to tell us this first sentence, I would have known to look at the rational numbers to order them from least to greatest. Well anyway, they tell us 1/2, negative five, three point three, zero, 21 over 12, negative five point five, and two and 1/8ths. So the easiest way to visualize this might just be to make a number line that's long enough that it actually can contain all of these numbers, and then we can think about how we can compare them. So let me just draw a huge number line over here. So, take up almost all the entire screen. I'll stick with, we have some negative numbers here, we go as low as negative five point five, and we have some positive numbers here, looks like we go as high as three point three. This thing is still a little less than two, so we go about as high as three point three, so I can put, I can safely I think put zero right here in the middle, I can go a little bit to the right since we have our negative numbers go more negative. So zero, and let's make this negative one, negative two, negative three, negative four, negative five, and well that should be enough. Negative five, and then in the positive direction we have, one, two, three, in the positive direction. And let's see if we can plot these. So, to start off, to start off let's look at 1/2. Where does 1/2 sit, so it sits, let me actually make the scale a little bit better. So this is one, two and three, and four. Alright, so let's start with 1/2. 1/2 is directly in-between zero and one, it is half of a whole. This right here would be one whole. This would be one whole, let me label that. This over here is one. So 1/2 is directly between zero and one. So 1/2 is gonna sit right over here. So that is, let me write that a little bit bigger, you probably have trouble reading that, Alright, one over two, which is also zero point five. So this is also zero point five, anyway that's where it sits. Then we have negative five. Negative five, well this is negative one, negative two, negative three, negative four, negative five. Negative five sits right over there. And then we have three point three. Positive three point three, I'll do that in blue. Positive three point three. So this is one, two, three, and then we want to do another point three. So point three is about a third of the way, a little less than a third of the way, it would be three point three three three forever, if it was a third of the way. So a third of the way, that looks like about right over here. This is three, this right over here would be three point three, let me label. What I'm gonna do is I'm gonna label the numbers on the number line up here so it's one, two, three, four, this is zero, negative one, negative two, negative three, negative four, negative five, and so on and so forth. And then we get to zero, which is one of the numbers that we've already written down. Zero is obviously right over there on the number line, so I'll just write this zero in orange to make it clear, it's this zero. Then we have 21 over 12, which is an improper fraction, and to think about where we should place that on the number line, to think about where to place it on the number line, let me do this in this blue color. To think about where to place this on the number line let's change it into a mixed number, makes it a little bit easier to visualize, at least for my brain. So 12 goes into 21, well it goes into it one time. One times 12 is 12. If you subtract you get a remainder of, well we could actually regroup here, or borrow, if you don't want to do this in your head, you would get nine, but let's do this. So if we borrow one from the two, the two becomes a one, this becomes 11, or we're really regrouping a 10. Anyway, 11 minus two is nine, one minus one is zero. So we have a remainder of nine. So this thing, written as a mixed number, 21 over 12 written as a mixed number is one and 9/12ths. You get one 12/12ths in there and then you get 9/12ths left over. So one and 9/12ths we can also write that, actually we could've simplified this right from the get go, cause both 21 and 12 are divisible by three, but now we can just divide nine, we can simplify 9/12ths, divide both the numerator and the denominator by three, we then get one and three over four, one and 3/4ths. And just to make it clear, I could have simplified this right from the get go, 21 divided by three, is equal to seven, and 12 divided by three, is equal to four. So this is the same thing as 7/4ths, and if you were to divide four into seven, four goes into seven one time, subtract, one times four is four, subtract to get a remainder of three, one and 3/4ths. So going back to where do we plot this? Well it is, it's one, and then we have 3/4ths, we're going to go three fourths of the way. This is half way, this is one fourths, two fourths, three fourths, would be right over there. So this is our 21 over 12, which is the same thing as 7/4ths, which is the same thing as one and 3/4ths. And then we have negative five point five. Negative five point five, I'll do that in magenta again, running out of colors. Negative five point five, well this is negative five, so negative five point five is going to be between negative five and negative six. So let me add negative six to our number line, right here just to make it clear. So let me go a little bit further, let's say that this is negative six. Negative six, and our number line will keep going to smaller values. Let me scroll to the left a little bit. Negative six, so if we go to negative five point five, it's smack dab in-between negative five and negative six. So this is negative five point five, right over there. And then finally we have two and 1/8ths. I'll do that in orange again, or I'll do it in blue. Two and 1/8ths, so it's two and then 1/8th. And so if we want to find the exact place we could divide this into eighths, this would be 4/8ths, this would be 2/8ths, and that would be 6/8ths, and then 1/8th would sit right over here. So that right over there is two and 1/8th. So we've actually plotted, as best as we could, the exact locations. You didn't have to plot the exact locations if you were just trying to order them, but it doesn't hurt to see exactly where they sit when we order them. So now we've essentially ordered them cause we stuck them all on this number line. The order is negative five point five is the smallest, then negative five, then a zero, and then positive 1/2, then 21 over 12, then two and 1/8th, and then three point three. And we're done.