Binary operators

Blasted things! In principle they are a shorthand used by math experts butin practice they are just something to hit kids with at high school.

Really all they are doing for you is developing your ability to read andunderstand abstract notation.

Here’s a concocted situation where a binary operator might be used.

Maybe I’m in the job of measuring triangles and use Pythagoras’s rulequite a lot. You know, the formula that associates the length of theslanting side with the lengths of the base and the height. Most peoplesee it written like this:

222 c b a

b

c

Here’s a concocted situation where a binary operator might be used.

Maybe I’m in the job of measuring triangles and use Pythagoras’s rulequite a lot. You know, the formula that associates the length of theslanting side with the lengths of the base and the height. Most peoplesee it written like this:

222 c b a

… although it would make a lot more sense if it was written like this:

222 s heightbaselant

Here’s a concocted situation where a binary operator might be used.

Maybe I’m in the job of measuring triangles and use Pythagoras’s rulequite a lot. You know, the formula that associates the length of theslanting side with the lengths of the base and the height. Most peoplesee it written like this:

222 c b a

… although it would make a lot more sense if it was written like this:

222 s heightbaselant

… and even more sense if it was written like this:

22 s heightbaselant

22 s heightbaselant

This could also be written in a more compact form (a kind of shorthand)like this:

height baseslant Δ

… where I have used the triangle symbol to represent the action of squaring both sides, adding them together and finding the square root.

I could have used any symbol on my keyboard; the triangle symbol seemed the most appropriate.

Okay, so I’ve created a binary operator called Δ. So what?

height baseslant Δ

Well, I have to show people (mathematically) what the triangle symbol actually does. So I write this definition in a book somewhere for people to read:

22Δ height base height base

Now that means that if somebody gives me the two short sides of a triangle(like 2 and 3), and asks me to get the third side, I can write the task down likethis:

3Δ2 lengthsideslanting

2

3

Now that means that if somebody gives me the two short sides of a triangle(like 2 and 3), and asks me to get the third side, I can write the task down likethis:

It seems to me there’s not much advantage to writing the formula like thisunless it’s going to be incorporated into an even bigger formula.

heightbase

heightθ

Δ sin

3Δ2 lengthsideslanting

Perhaps someone doing trigonometrymight like to define the sine formula interms of this new triangle symbol:

It looks more complicated to folks who are seeing it for the first time, but afellow who has to use trigonometry frequently might find it convenientbecause he doesn’t have to write so many symbols.

heightbase

heightθ

Δ sin

It looks more complicated to folks who are seeing it for the first time, but afellow who has to use trigonometry frequently might find it convenientbecause he doesn’t have to write so many symbols.

heightbase

heightθ

Δ sin

He might even be able to create a whole new kind of algebra around it.

For example, what happens when we put two of these things together?

??? Δ Δ cba

It looks more complicated to folks who are seeing it for the first time, but afellow who has to use trigonometry frequently might find it convenientbecause he doesn’t have to write so many symbols.

He might even be able to create a whole new kind of algebra around it.

For example, what happens when we put two of these things together?

??? Δ Δ cba

Can it be rearranged to equal ? Δ Δ cba

And if so, what would that actually mean?

22 Δ baba

22

22 Δ Δ cbacba Therefore

cba

cba

Δ Δ

222

Which means that it doesn’t matter where you put the brackets.The answer will be the same.

The only way to work this out would be togo back to basic algebra and test the idea out.

An academic question such as this would be of intense academic interest to(ahem) academics and virtually no-one else. But it shows that newmathematical language can be created wherever there is a need for it, and thatthe behaviour of that new symbol can be investigated, sometimes withsurprising and useful results (to an academic).

22

22 Δ Δ cbacba

cba

cba

Δ Δ

222

a

b

c

In the case of ‘a Δ b Δ c’, it turns out that what we are doing is moving trianglesinto the third dimension. ‘a Δ b Δ c’ gives us the length of the green line in thepicture I have drawn.

a

b

c

Using mathematics such as this, wecan imagine our way into universesthat have more dimensions than ours,and even construct mathematicalshapes in these places.

In the case of ‘a Δ b Δ c’, it turns out that what we are doing is moving trianglesinto the third dimension. ‘a Δ b Δ c’ gives us the length of the green line in thepicture I have drawn.

So is it possible to go one stage further, as in ‘a Δ b Δ c Δ d’ ?

Mathematically, yes it’s possible. But physically? I’m not so sure.

Looking back into history, it seems to me that the first binary operator ever tobe invented was the multiplication symbol.

‘2x6’ is a shorthand for “add two to itself six times”

Of course, the plus symbol and the minus symbol precede it, but these are thebasic definitions upon which all other mathematical symbols are constructed.‘Times’ is surely the first mathematical shorthand to be constructed out ofsomething simpler, with ‘Divide’ as a close second.

Some binary operators are reversible,in the sense that 2 x 15 is the same as 15 x 2.

That’s pretty neat because we can view the addition of fifteen 2s as merely theaddition of 2 fifteens. Even better, we can conceive of something called thetimes table, which gives us more computational power when we commit it tomemory.

Perhaps this is what those thick-spectacled academics are doing when theyinvestigate the properties of a new mathematical symbol. If it can be shownthat ‘a Δ b’ is the same as ‘b Δ a’, or that ‘a Δ a’ is something simple, thenmaybe one day our kids will be memorising a triangle times table in the sameway that we memorise the multiplication times table. And who knows wherethat will lead?

In the case of the triangle operator that I havedefined, simply for this lesson, here are someinteresting properties that I’ve observed:

aaa 2 Δ

aaaaaa 3 Δ 2 Δ Δ

anaaa (n times) Δ ... Δ

In the case of the triangle operator that I havedefined, simply for this lesson, here are someinteresting properties that I’ve observed:

aaa 2 Δ

aaaaaa 3 Δ 2 Δ Δ

anaaa (n times) Δ ... Δ

Perhaps I should create a new symbol now torepresent what happens when the trianglesymbol is applied to itself n times. anaa Δ n

For example, does ‘n’ have to be a positive integer?

What happens to when n is a negative number?

- or a fraction?- or zero?- or even a complex number?

Now the invention of this new mathematical language is stimulating meto consider questions I would never have thought of asking beforehand.

aa Δ n

Once again the answer to a question is bigger than the question,and is leading to more questions.

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