This week we started looking at equivalence. The first question that was asked was, undoubtedly, “what is the first thing that comes to mind when you think of equivalence?”
Well, as you may or may not know, I’m taking a pseudo-vow-of-silence this week. This means that I did not answer the question, though I did answer it in my head. Naturally, the first thing that came into my head was logical equivalence, an expression in propositional logic that translates into English as “if and only if” and is shown by what can be explained as a long “equals” sign with arrowheads on each side pointing away from each other. For example P is logically equivalent to Q (I don’t have the symbol on my keyboard because I’m surprisingly normal in that regard) means that P and Q must have the same logical content, which is where we get all of our logical equivalence laws from, such as De Morgan’s laws, associativity, commutativity, double negation and so on and so forth. However, this is not an article on logical equivalence, this is an article on mathematical equivalence.
Which is a very similar concept. Now, we were not looking at anything complex; the course that I am on is designed for people who have not done A-level mathematics, which still comes as a surprise to me, much like how my course mates on Computer Science as an undergraduate hadn’t done A-level mathematics, despite it being listed as necessary on the prospectus. We were looking at basic equivalences. I feel an example is warranted:
I go into a super market and buy something for half price, because sales, offers and reduced to clear items are how I live. I get to the checkout and get chatting with the cashier and comment on how I only paid 50% of the original price of the item. I then decide to work out what my saving on the item was (because I didn’t look at the reduced price, only the original and the fact that it was half price. I get impulsive around food…). I type into my calculator the original price (which we’ll say was £3.50) and multiply it by 0.5.
Deconstructing the example we have the concept of half (1/2), a percentage (50%) and a decimal value (0.5). All of which are equivalent (but not equal!).
These are not equal because whilst you can write 1/2 = 0.5, you can’t write 1/2 = 50%. 50% of what? 50% of 1, sure. What about 50% of 10564? That is definitely not equal to 1/2! 50% requires context, so it is not necessarily equal to 0.5 or 1/2, however it is equivalent. We, or at least I, use equivalence without considering that we are using it. For example, if I were working out what 25% of a number is, I will default to a quarter and just divide it by 4. If I wanted to work out what 43.8% of a number (without using my calculator) I will, again start using fractions which are equivalent to what I need (43.8 -> 43 4/5, 43.8% -> 43 4/5 / 100) as I find them easier to use than playing about with decimal numbers and place value.
The largest area in the maths syllabus where equivalence comes up is, with no surprise from me, in fractions. Equivalent fractions are a large topic. Then again, it’s important to be able to spot equivalent fractions or work them out. From a basic point of view, it means that you don’t have to deal with such out of control numbers. What would you rather deal with? 1/3 or 17/51? I know that I am lazy and love to keep my life simple, so I would take 1/3. You can have 17/51 if you really want. It’s yours. Go nuts.
I came up with a way of explaining why simplifying, or inversely simplifying if you need to match denominators for an addition or subtraction, works the way it does. So, first a general question to you, the reader. What happens when you multiply any number by 1?
I really, really hope that you said “it stays the same” or “nothing changes with the original number” because, when considering equivalence, that is totally true. Multiply anything by 1 and it will not change. However, consider this: take 1/2 + 1/8. We need to get 1/2 in terms of eighths (that’s a really odd word to spell…) so we multiply it by 1.
“But hang on…” I hear you say, “1/2 multiplied by 1 is still 1/2.” You are totally right about that, but that’s what you get if you multiply 1/2 by 1/1. 1/1 is just a fancy fractional way of writing 1. So is 2/2. And 3/3. And even 1/2 / 1/2. So, to get 1/2 in terms of eighths, we multiply it by 1, otherwise written as 4/4 (as we divide the larger denominator by the smaller one to know what number we need to multiply the smaller one by to get the larger number. Remember, division is reverse multiplication). 1/2 * 4/4 = 1*4 / 2*4 = 4/8. Therefore we can write 1/2 as 4/8 and thus add 1/8 to it to give us 5/8. Not 5/16. In addition and subtraction we do nothing to the denominator once we have made them the same. Nothing. We don’t touch them. It’s like they suddenly got a bad flu and we want nothing to do with them. Or it peed itself and now has a fairly anti-social smell of urine about it.
Similarly when we’re simplifying fractions all we are doing is multiplying by 1. You find a number that is a factor of both the numerator and denominator (either by deconstructing both to their prime factors or just seeing it/testing numbers) and then make a fraction of it as a unit fraction over itself. This is incredibly awkward to explain without paper and pencil, so I will give an example. With 17/51, we can see that 17 is a common factor of both numbers (17 * 1 and 17 * 3). So to reduce it we simply multiply it (or divide it. Multiplying by 1 and dividing by 1 give the same answer!) 1/17 / 1/17. 17/51 * (1/17 / 1/17) = (17 * 1/17) / (51 * 1/17) = 17/17 / 51/17 = 1/3.
Of course most of us do this in a much more simple looking way, but the roundabout explanation of it is this. And it all revolves around the idea of equivalence. Equivalence to solve equivalence.
Similarly we can use this concept to explain how to divide fractions without simply saying “you turn the right hand one upside down and multiply!” by again changing the question to “dividing by 1.” What we have to remember is that a fraction is just another way of saying the numerator divided by the denominator; we even use / to represent this when typing on a keyboard as I am right now. Therefore 1/3 divided by 5/12 is the same as 1/3 / 5/12, where 1/3 is the numerator and 5/12 is the denominator. This concept of fractions as a part of a fraction is a very important concept to understand, in my opinion, as it provides a very strong base from which to springboard into fractional arithmetic and understanding. So, now that we have our fraction built up of fractions, how do we simplify it? Well, we want the denominator to equal 1 and the tools at our disposal are multiplication and division, as addition and subtraction will change the number (or involve adding or subtracting 0, which is not useful), unlike multiplying or dividing by 1! So, breaking it down… What do we multiply 5/12 by to get 1? Well, if we multiply it by 12 we will get 5, which is not what we want. So how do we get from 5 to 1 using multiplication or division? Well, we divide by 5 as 5/5 = 1. So, we’ve got there! How did we get there? We multiplied by 12 and divided by 5. So we multiplied by 12 divided by 5, or 12/5. However, we need to remember the original question and the fact that we’re using the idea of multiplying by 1. We’re multiplying (or dividing) the original fraction built of fractions by 1 as to not change what the original number is, so whatever we do to the bottom (denominator) we have to do to the top (numerator). So we’ve multiplied the bottom (5/12) by 12/5 to get 1, so we must multiply the top (1/3) by 12/5 as well. This gives us (1*12 / 3*5) / 1 = 12/15 / 1 = 4/5 / 1.
4/5 divided by 1 is simply 4/5. Acquired because of 2 simple concepts. x*1 = x and equivalence (2/2 = 5/5 = 1043/1043 = 1)
Now let us look at what happens if you simply turn the right hand fraction upside down. 1/3 divided by 5/12 becomes 1/3 * 12/5, the exact calculation we were doing for our numerator in the previous paragraph. 1/3 * 5/12 = 12/15 = 4/5. We reach the same answer, but skip the excessive number manipulation.
Also, to highlight my point on fractions being easier to use than decimal numbers: try doing 1/3 / 5/12 as decimal numbers without a calculator. So that’s… 1.333333333… recurring 3s, divided by 0.4166666666666666666… recurring 6s. I’ll wait over here and continue to sing and dance along to Bad Romance by Lady Gaga. Take your time. You’ll need it.
You remember when I said that I’d wait for you to calculate it?
(And I really wanted to make an Arnold Schwarzenegger reference).
There isn’t actually much more I wanted to write about in this article. I wanted to touch base on equivalence and show that it’s largely taken for granted. I also wanted to show the power of multiplication and division by 1 using equivalence to change what it looks like you are doing. In algebra we write x + 3y = 8z, not 1x + 3y = 8z, because x is the same thing as 1*x. Also, to finish on an interesting observation… Take 36 divided by 9. We know this is 4, yes? Now, multiply 36 and 9 by 218 and divide them again. The answer is still 4.
I wonder if that’s been named yet. I bet it has. My quest to have a mathematical concept named after me is, as usual, still not being completed.