Zero and Negative Infinity

Date: 06/02/2003 at 19:42:32
From: Sree Dharani
Subject: Smallest number - Zero or negative infinity

What is the smallest number? Zero or negative infinity?

I was able to sort of explain the concept of infinity to my almost 
4-year-old as a very large number. Her next question was 'what is the 
opposite of infinity?' 

I am getting confused between the concept of 'less' and 'small' when 
it applies to numbers.

I would also like a real world example of negative infinity that I 
can use to explain it to my daughter.

Date: 06/03/2003 at 14:34:47
From: Doctor Achilles
Subject: Re: smallest number - Zero or negative infinity

Hi Sree,

Thanks for writing to Dr. Math.

It's great that you're taking time to talk about high-level math 
concepts with your child. It's often best to only teach one hard 
concept at a time, like infinity, and let that one sink in before 
teaching something else that is also difficult, like negative numbers.

Zero is the smallest number. Infinity isn't a number; rather, it's 
the concept of something larger than any number, but for a 4-year-old 
it can suffice to say it's the largest number to get the idea across 
as a first pass.)

What follows is not necessarily an explanation for a 4-year-old:

The difference between "less" and "small" is very difficult 
(especially to explain to a 4-year-old. I think of them in terms of a 
number line. The number line is a line that starts at zero and goes to 
negative infinity on the left and to positive infinity on the right.  
Something like this:


(to neg inf)<--------------------|-------------------->(to pos inf)
           ...-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6 ...

If you're comparing two numbers, you can say that the one farther to 
the left on the number line is "less" and the one farther to the right 
is "greater."  "Less" and "greater" are concerned with *position*: 
where the number lies on the line.

"Small" and "big" on the other hand are concerned not with position, 
but with *size*. Zero is defined as the smallest number. The closer 
something is to zero, the smaller it is. Size is a way to talk about 
"absolute value."  The absolute value of a positive number is just 
that number, and the absolute value of a negative number is just the 
inverse of it (so the absolute value of -4 is 4).

I hope this helps.  If you have other questions or you'd like to talk 
about this some more, please write back.

>I would also like a real world example of negative infinity that I 
>can use to explain it to my daughter.

Regarding this question, I don't think I can help because I am at a 
loss to come up with an example of negative numbers or of positive 
infinity which would work for a 4-year-old (and I can't even come up 
with an example of negative infinity for anyone, offhand.

Best wishes for you and your child in working through tough math 
concepts.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/03/2003 at 15:02:45
From: Sree Dharani
Subject: Thank you (Smallest number - Zero or  negative infinity)

Thanks Dr. Achilles.

That was a speedy response and has provided me the relevant direction 
to answer my daughter. I discovered this site accidentally and am 
happy for that discovery. I am sure I will visit this site many more 
times in future.

One of my colleagues was suggesting the 'debt of U.S.' as a closest 
example of negative infinity he could think of.

I was thinking of explaining negative infinity relative to current 
time.  The time when the universe was created is negative infinity 
seconds from the current time.

Date: 06/03/2003 at 17:10:12
From: Doctor Peterson
Subject: Re: Smallest number - Zero or  negative infinity

Hi, Sree.

Properly speaking, infinity is not "a very large number," as you have 
said; I would say it is BEYOND any number you can think of. In fact, 
infinity really has nothing to do with numbers; the main idea (if you 
look at the Latin root of the word) is "endlessness," reflecting the 
fact that the number line goes on forever. Too many children get a 
wrong idea of what infinity is, from attempts to give them an 
explanation at this level, which they later have to unlearn.

So I would avoid connecting infinity with large numbers, and 
certainly with actual large numbers, even as a joke. Rather, I would 
say that infinity is "as far up as anyone can imagine, and then 
some." And negative infinity is "as far DOWN as you can imagine, and 
then some."

I don't think you've told us how well your daughter understands the 
idea of negative numbers, or of positive infinity; I think the best 
early introduction to the former is on the number line, with the idea 
of "numbers" to the left of zero. In my version above, I avoided 
"left" and "right" and used "up" and "down" because the latter seem 
more intuitively positive and negative than the latter, for someone 
not yet accustomed to the conventions of the number line. 
Representing negative numbers as used in counting (as in your example 
of a debt) seems like a later concept, not the best way to introduce 
the concept.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/