Order of Operations, Part Two

OK, now we have learned about the opposites of the operations, does that make a difference to our rules about the order we do them in?

Sadly, and predictably, yes.

What is three minus two plus one? Is it three minus two (one) plus one (two)? Or is it two plus one (three) taken away from three (zero)?

Oh, bugger. But hang on, didn't I say that subtracting a number was the same as adding the negative of that number? So what if we just pretend, between us, that there is no such thing as subtraction. Lets concentrate on addition.

So what is three plus negative two plus one? Is it three plus negative two (one) plus one (two)? Or is it negative two plus one (negative one) plus three (two)? Hey - it's both! Yes, by getting rid of subtraction altogether the problem of the order goes away.

So what about multiplication? What is two multiplied by three divided by four? Is it two multiplied by three (six) divided by four (six fourths, or one and a half)? Or is it three divided by four (three quarters) multiplied by two (six quarters or one and a half)? Hey - it doesn't matter. Or does it? What about two divided by three times four. Is that two divided by three (two thirds) multiplied by four (eight thirds)? Or do you take three multiplied by four (twelve) and divide two by it (two twelfths, or one sixth)? Oh bugger, a sixth is not the same as eight thirds.

So the best idea is again to forget about division and instead stick with multiplication by the inverse. So for two divided by three multiplied by four, we can say two multiplied by a third (two thirds) multiplied by four (eight thirds), or a third multiplied by four (four thirds) multiplied by two (eight thirds)! Check.

Because we are now just adding (negatives) and multiplying (inverses) again we can stick to the same ordering conventions we decided upon before.

What about the opposite of exponents? Hmm, tricky. What is the square root of four plus one. Does that mean the square root of four (two) plus one (three)? Or does it mean the square root of five? Again, if you stick to square roots written as fractional exponents it clears this all up. You would say four to the power of a half plus one. And then ... bugger. Is that one plus a half (one and a half) to which you then raise four? Well, no because we have already decided that you do your exponents stuff first, and unless it is in brackets it is definitely four to the power of a half (two) plus one.

So, the opposites of addition, multiplication and exponents can cause problems with the ordering of operations, but if you just ignore the opposites and add, multiply or exponent (if that's a verb) by the new numbers we discovered then you are going to be OK with Brackets -> Exponents -> Multiplication -> Addition.

Now we are happily done with the different kinds of mathematical operations that we need to know about for this little adventure. It is time to get back on track with the subject at hand. We've looked at zero and one, so it is now time for one of those funny symbols.