Let’s take a look at comparing and ordering integers.
Here’s the key to ordering and comparing integers. When two numbers are graphed on a number line, the number to the left is always less than the number on the right. The number to the right is always greater than the number to the left. Here negative two is to the left of zero. We would write it in this way, negative two is less than zero. Here’s another case. The one is to the right of negative three. We would write one is greater than negative three because the one is to the right.
Let’s check out some examples like this one, negative twelve is less than negative thirteen, true or false? Let’s use a number line to think through this. Here’s what it looks like when we count with our negative integers. So I extended out our number line and added negative twelve and negative thirteen in their respective places. So the rule is, the value to the right of the other value on the number line is larger. In other words, as we move to the right on a number line, the values are getting bigger. In this case, negative twelve falls to the right of negative thirteen, which makes negative twelve a larger value than negative thirteen. The problem in our example lists negative twelve as less than negative thirteen, which makes this statement false. Negative twelve is not less than negative thirteen.
Here’s another example, example two. Fill in the box with greater than, less than, or equal to. Negative ninety-six is greater than, less than, or equal to the absolute value of negative ninety-six. Remember, absolute value is a numbers distance from zero on a number line. Before we move forward in answering this question, we need to figure out what the absolute value of negative ninety-six is. The distance from negative ninety-six to zero on a number line is ninety-six units. Now we’re looking at something like this negative ninety-six compared to ninety-six. Back to our number line again, as we moved to the right of the number line, our values are increasing, so we can say that negative ninety-six is less than ninety-six. Or in the original example, negative ninety-six is less than the absolute value of negative ninety-six.
We don’t just want to be able to compare two different values though. Sometimes we want to order values. When we’re ordering values, we’re taking a whole set and we’re putting them in order, from least to greatest, or from greatest to least.
Example three is asking us, order the following from least to greatest. Negative seven, nine, eighteen, fourteen, negative eleven, negative ten. Back to our trusty number line, we’ll put our zero in the middle. Okay. Now I’ve just divided up the values that we’re working with and I put the negatives on the left side of zero, because I know that’s where they’re going. And I put all of the positive values on the right side of zero, because I know that’s where those will go. Ordering the positive values will not be difficult. Nine is the smallest of the three, next comes fourteen, and then eighteen. Those are good to go.
But we have to be a little bit more careful when we’re working with negative values. We can actually use absolute value to help us do this. We know that the absolute value of a number is its distance from zero. So as I’m placing negative seven, negative eleven and negative ten on the number line, first I’m gonna find their absolute value. Negative seven would fall seven units away from zero on a number line. Negative ten has an absolute value of ten, so it’s the next number in our list. It falls ten units from zero on the number line. And finally, our negative eleven falls eleven units from zero on the number line.
But our work here is not done. I said a few times as we move from left to right on a number line, our values are getting larger. This is exactly how we number least to greatest. We start at the furthest left and we move to the furthest right. That’s what the list would look like, ordered from least to greatest.
But what if we change the problem, and we try to order them from greatest to least. The opposite would be true here. If we start to the far right and move to the left, the values are getting smaller. And when you want to move from the greatest to the least, you would then wanna look at the right and then move to the left. This is what an ordered list moving from greatest to least would look like, eighteen, fourteen, nine, negative seven, negative ten, and then negative fourteen being the smallest.
Here’s our last example. A helicopter is hovering at a height of six hundred and forty-seven feet, a diving bell is at negative five hundred and sixty-seven feet. Which one is farther from sea level? Our question, which one is farther from sea level, the two pieces of information that were given. But we also have to remember, what would it mean to be at sea level. What number value is sea level? If something is directly at sea level, its zero feet. In our case, let’s put our helicopter here at six hundred and forty-seven feet, and we put the diving bell here. Our question is asking which one is farther from sea level. Another way we could say this is, which one is further from zero. The diving bell is five hundred and sixty-seven feet below sea level, or five hundred and sixty-seven feet from sea level. And our helicopter is six hundred and forty-seven feet from sea level, or from zero. Six hundred and forty-seven is greater than five hundred and sixty-seven. Did you catch that we’re actually using absolute value here? We found the absolute value of negative five hundred and sixty-seven. The final answer here would be the helicopter. The helicopter is farther away from sea level. Good luck! And see you next time.