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/ |
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