What's smaller than the smallest

You have probably already noticed that the negative numbers on the number line look like a mirror image of the positive numbers. But which negative number is larger and which smaller? Read through the following discussion on this topic and consider who you agree more with. Do you think Peter is right or do you think Susanne is right? Peter: If I owe 5 euros, then I owe more than if I only borrowed 3 euros. So -5 must be greater than -3.

Susanne: But we learned beforehand that debts are not given in negative numbers! Look, if I have -5 euros in my account, I have a lower balance than if I only have -2 euros in my account. Therefore -5 must be less than -2.

Peter: Okay, in that case you might be right. But what if it is -10 C? Then it is more cold than at -2 C. Then I'm right again.

Susanne: But here, too, it has to be the case that -10 C is a lower temperature than -2 C, right? So -10 should be less than -2 again.

But which of the two is right now? Take a look at the following picture and consider which side is the representation of Peter's opinion and which side is the representation of Susanne's opinion.

Peter and Susanne still don't agree who is ultimately right. It works Marianne into the discussion. She says: "You surely won't deny that 4 is less than 5, will you?" Marianne writes the inequality on the blackboard.

4 <>

Then she says: "If we subtract 3 on both sides, the inequality is still true."

So she writes on the board:

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"If we subtract 3 again, this is what comes out."

-2 <>

"So -2 is definitely less than -1."

Marianne is right. The order of the negative numbers is exactly the same as that of the positive (and NOT MIRRORED!), so a number (a) that is further to the left on the number line than another number (b) is smaller (i.e. a <>

Task: Download the following file and print out the document. Do the tasks and stick the exercise sheet in your school exercise book.

Inequalities.pdf

So, now you've done quite a bit. You now know the extended number line and the extended coordinate system, you can draw points in it and order negative numbers according to their size. You have already done a large part! :-)