True or False PART II

The other day I asked you all a true or false question:

I brought this up at school today and it created quite a bit of discussion.


True or False:

If x^a = x^b  then a = b. 

Let me know your answer in the comments below!

I'll do a full post on this once I have some answers.

Emma x x x 

 Thank you to everyone who replied. As well as asking on my blog, I asked my students, my colleagues, and some non-mathematical people, including my dad. There are a few interesting things to discuss regarding this question. 


The "logic" of the question


The question I posed contained no existential or universal quantifiers. However, some people reading the question assumed that there were implied quantifiers which completely changed the meaning of the question. For example:


If x^a = x^b  for all values of x then a = b. Is this statement true or false? I think it's true.


The other thing some people confused was the logic of the phrase "if... then...". I would normally use an implication symbol instead but a) I don't know how to type that (my html is an fml) and b) I thought that it's likely not everyone would know its meaning. What surprised me was that even phrasing it this way, there were some people who misunderstood. I received some responses along the lines of "it can't be false because it works for [example]". They didn't understand that for this statement to be false only one example has to fail. This sort of logic isn't taught until A level maths. 


The types of number allowed in the question


 It is obvious that if we restrict x, a and b to natural numbers excluding 0 and 1, the statement is true. A lot of people, seeing the letters a and b, assumed they had to be integers. Just for fun, I asked the question to different people using different letters. When I used y and z, they were less likely to assume they were integers than when I used n and m. 

When x = 0 or 1, a and b can be anything*, rendering my question trivial. So to make things more fun, I told some people that they weren't allowed to use zero or one. This means to find a counterexample you have to go into complex numbers.

 Complex numbers and e

The question had first occurred to me when I was teaching a Further Pure 2 lesson, aand we had to solve the equation e³ = e^z which prompted one of my students to say, surely the answer's just z = 3? I had already gone through the solution on the board and we had come to the answer which by the way is z = 3 + 2kπj (for integer k). My student didn't believe this could be true, and I even started doubting it myself. I couldn't see how z could be anything other than 3. e is just a number, after all. It's kind of amazing when you think about it. That is what is so awesome about complex numbers, they always manage to surprise you. The reason why it works is because you can write

e^3+2kπj  as   e³  x e^2kπj   and it can be shown that e^2kπj = 1.

When I told somebody that this was the type of counterexample I was looking for, they protested, saying that I couldn't use e because e is a function. I replied that e wsan't a function, it was a number, 2.7182818... a bit like pi. They still weren't happy with this, and changed their track. x stands for a variable, but e isn't a variable, it's a constant. I found this very hard to explain: x is a variable and e is one instance of that variable. That particular instance doesn't work, so the statement is false. The person still wasn't convinced.

So, a simple question, but it generated a lot of discussion. It highlighted to me how many people (including maths teachers) have misconceptions with the whole idea of proof and logic and  numbers. 

Wow, this was a pretty heavy post for the last day of the school year! 

Fellow teachers, enjoy your freedom!

Emma x x x

PS If you're wondering why I call it j instead of i, it's because at my school we follow the MEI A level programme, and their links with engineering means it makes sense to use the letter preferred by those in industry. I also followed this syllabus when I studied A level myself, so j is what I grew up with. It was so weird at uni where everyone called it i! 

PPS I now have a contents page! Check it out!

*OK, not completely true, depending on how you define zero to the power of zero.