Ordering rational numbers
I've got five numbers here, and the goal is to order them from least to greatest. And it might be obvious to you that all five of these numbers are negative numbers. So let's just think about which of these is the greatest number. And you might be tempted to say oh, well, you know, if these were all positive, 40 would be the greatest. So you might be tempted to say negative 40 is the greatest. But you have to realize what the negative is telling you, and to think about that. Think about if these were your dollar amounts in your bank account. Would you rather have negative $40 in your bank account or negative $7 in your bank account? Negative 40 means that you owe the bank $40. So it's 40 less than having nothing. Negative 7 would mean that you owe the bank only $7. So it's actually the case that negative 40 is less than negative 7. And out of all of these numbers, it is the least of all of them. So negative 40 is the very least that you could do. That's the least amount of money that you would want out of these comparisons in your bank account, or the smallest amount of money. You owe the bank $40. Not only do you have nothing, you owe the bank $40. Then the next smallest number would be negative 30. And then the next smallest after that would be negative 25. And then the next smallest after that would be negative 10. And then finally, the greatest of all of these numbers-- and I will do it in pink-- the greatest of all of these numbers is negative 7. And if it's still not obvious to you, you could also think of them in temperature terms. Which of these is the coldest temperature, whether you're talking about Celsius or Fahrenheit? Negative 40 would be the coldest temperature. Negative 7 would be the warmest temperature. There would be the most heat in the air at negative 7. Another way to think about it is we could draw a number line. So let's draw a number line right over here. If this is a zero, and we could put maybe positive 7 up here. Positive 7 isn't one of these numbers, but then we could plot all the other numbers. So this right here might be negative 7. If that's negative 7, then maybe this right over here would be negative 10. Notice we go further and further to the left of zero. And then we go a little bit further to the left. We get negative 25. A little bit further to the left than that is negative 30. And then you go even further to the left and you get to negative 40. So if you think of it this way, the least of these numbers is the one that's furthest to the left on the number line, and the greatest is the one that's furthest to the right.