What Is The Largest Number Smaller Than Five?

When my sons were in 11th grade at Dover/Sherborn High School, their AP BC Calculus teacher, Mr. Bridger, used to start every year with a question: “What Is The Largest Number Smaller Than Five?”

My boys would come home from day one with their heads spinning from the rich philosophical discussion that ensued. I decided to kick off 2019 Mathletes with the same question at all grade levels.

It is fascinating that their answers are dependent on their knowledge of classifying numbers:

• natural numbers {1, 2, 3, 4, …}, 
• whole numbers {0, 1, 2, 3, …}, 
• integers numbers {…, -3, -2, -1, 0, 1, 2, 3, …}, 
• rational numbers {all fractions that can be written as a/b where a is an integer and b is a natural number; ex. 4, 1.2, 1/7}, and 
• irrational numbers {decimal doesn’t repeat or end; ex. square root of 2, √3, √5, pi=3.141592653589793….}. 

My K/1st graders would say 4; my 2/3rd graders would say 4 1/2; my 4/5th graders would say 4.9 or 4.99999…; and my 6th graders would start at 4..9 repeating, and then the fun would begin.

First, the question is what is a number? 

K/1st graders have a very good understanding of the natural or counting numbers or positive integers. So, of course, their perspective is one less than 5, so they say, “4.”

2/3rd graders knew that there is something between 4 and 5 and so they correctly split the difference with 4 1/2. 

4/5th graders cleverly know that the way to get closest to 5 is to have 4.9 or 4.99999 repeating. But when I asked them if 4.999… is equal to 5, that is when the intellectual debate gets heavy. 

With my older Mathletes I was able to prove that 4.999… is equal to 5. Then I asked them the question again: What Is The Largest Number Smaller Than Five? Now, they responded, maybe “there is none.” See the attached pdf for the proof. 4.9… cannot be both less than 5 and equal to 5.

Some of them were not convinced, so I said let’s do a proof called reductio ad absurdum; this is Latin which means “reduce to the absurd.” Here it goes:

1. Assume that there is a largest number smaller than 5; call it x.

1. Take the average of x and 5 and call it y, which would be:        y = (x + 5)/2          

3.      y is smaller than 5 because it is the average of x and 5 and we assumed that x is less than 5.

4.    y is bigger than x because it is the average of x and a number 5, that is bigger than x.

5.    How can this be? In step 1 we said that x was the largest number less than 5 and now we are saying that y is greater than x and less than 5. This is impossible.

6.     Our original assumption has led to an impossibility, therefore there cannot be a largest number smaller than 5.

Here is how reductio ad absurdum works:

1. assume A is false
2. deduce an impossibility
3. say that A must therefore be true

Then the children started talking about infinity since there is always a number between any other two numbers. So whatever number you choose as your largest number less than 5; call it x, there will always be something bigger than your x and less than 5. 

With the younger set, we used number lines to show that there is always something in between any two numbers.

I asked some of the groups to find the number on a number line half way between 4 and 5, which is 4 1/2. Then, half way between 4 1/2 and 5, which is 4 3/4; then 4 7/8; 4 15/16; 4 31/32. Now continue infinitely which also proves that there is no largest number less than 5.

Walk half way from one end of the room to the other. Then half way from that point to the wall, then half way again, and so on. Of course, physically, we appear to hit the wall after about 6 or 7 half-walks but mathematically, we can never reach the wall.

I recently read The Fault in Our Stars by John Green, now a major motion picture. One of the ideas that resonates with Hazel, the 16-year-old narrator of the story, is the idea that “some infinities are bigger than other infinities.”

“There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities…. I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.”

The sentiment is lovely but mathematically inaccurate. One of the most mind-blowing facts a young mathematician learns is that, in a specific, rigorous way, there are exactly as many numbers between 0 and 1 as there are between 0 and 2, 0 and a million, or even in the entire set of real numbers! Don’t worry, it’s natural to feel dubious about that. It seems impossible that a set could be the “same size” as a set that contains it plus some other stuff! But that’s one of the marvelous mysteries of infinity.

This week, all I want them to do is to ASK AS MANY PEOPLE AS POSSIBLE (FAMILY, FRIENDS, TEACHERS): "WHAT IS THE LARGEST NUMBER LESS THAN FIVE?" AND TALK THEM THROUGH THEIR ANSWER AND WHAT YOU LEARNED IN CLASS.

Help them record 10 numbers in each number category (natural, whole, and think about how many numbers are between each consecutive whole number (how many numbers are between 0 and 1 or 1 and 2 or 1 and 100.

The younger Mathletes should stay away from irrational numbers unless they have been exposed to pi. Even my 3rd graders will know a few fractions, which are rational numbers that are not whole numbers.

They should bring their work to class next week and share how their conversations went when asking people the question: What is the Largest Number Less Than Five?