Is One a Triangular Number?

Date: 03/22/2004 at 06:42:42
From: Hayden
Subject: Triangular numbers

Is the number one (1) a triangular number?  Triangular numbers have 
that name because, if drawn as dots they can form a triangle.  But 1 
is just a single dot, so it can't be a triangular number, can it???

My Maths teacher marked a question in my test (triangular numbers
between 1 and 9 inclusive) wrong because I left out 1.  Both my maths
tutor and I disagree with him.  Who is correct?  Why?

Date: 03/22/2004 at 12:34:12
From: Doctor Peterson
Subject: Re: Triangular numbers

Hi, Hayden.

The teacher is right, as you'll see from the definition of triangular 
numbers here:

  Number glossary
    http://mathforum.org/dr.math/faq/faq.number.glossary.html 

We commonly do this in mathematics: we define a term initially based 
on an observation (such as that certain numbers of dots can form a 
triangle), then extend the definition a little more broadly (in this 
case, as a number that can be written as a sum of consecutive numbers 
starting at 1), and find that the new definition includes a case that 
is "on the edge" of this definition, namely when there is only one 
row of dots, so we are only adding one "consecutive number", 1 itself. 
It doesn't really look like a triangle (though you can stretch your 
mind and see it that way!), but it does continue the pattern of 
triangular numbers and gives a good starting point for the sequence.

The same is true of square numbers, which geometrically are those that 
form a square:

  o   o o   o o o
      o o   o o o
            o o o

Note that 1 is a square number (because it is 1 times itself) even 
though it's hard to picture one dot as a square.

What you have to do is to look at the actual definition of a concept 
like this, rather than at the original idea that motivated it.

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

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