Name: Jewel

Who is asking: Teacher
Level: Elementary

If zero means nothing or an empty set, then how can a number be less than zero, as in negative number? My understanding of negative numbers is distance from a set point in a given direction. Thus having a negative and positive side of a line is arbritary and is related to displacement rather than value. Am I in error?

Hi Jewel,

"Nothing" or "none" are words we sometimes use when we mean zero. "How many apples are in a bag of oranges?" might invoke the answer "There are none." This is a situation that arises when we are counting, and dealing with the counting numbers. The counting numbers are

0, 1, 2, 3, ...
There are no negative counting numbers. If you are dealing with temperature, and the thermometer reads zero, I doubt if you would say there is no temperature or the temperature is nothing. If the scale reads -3o F it's not three degrees less than nothing.

Negative temperature fits more closely to your "understanding of negative numbers is distance from a set point in a given direction". The zero is a somewhat arbitrary point on the temperature line and positive and negative temperatures do measure "displacement". However the negative is very important. Were I live and we see winter days with a temperature of -30o C (-22o F) we know that is quite different from 30o C (86o F).

The zero from which you measure positively and negatively might not be as arbitrary as the zero on the temperature scale. Many young couples buy a car and a house and find that their net worth is negative, that is they owe more than the value of their assets.

Negative numbers are real mathematical objects and serve a valuable purpose.

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