Articles of education

Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock

Fifty minutes ago it was four times as many minutes past three o’clock. How many minutes is it to six o’clock..? I have got the solution online but have doubts in it : There are 180 minutes between 3 o’clock and 6 o’clock. Call x the number of minutes to 6 o’clock. Then it is […]

At what education level is this “simple” problem adequate?

Given the following problem: Alice, Bob and Carl stand on a straight line. Alice is one rod away from Bob. Dana stands one rod away from both Bob and Carl. Carl is as far from Alice as Alice is from Dana. How far can Alice be from Carl? Answer using a well-formed sentence, with any […]

Why learn abstract mathematics? What is the point?

I’m a mathematics college lecturer and have an mphil degree in the subject. But I often wonder why I’m learning this senior undergrad level mathematics—analysis, topology, functional analysis, abstract algebra etc. Whatever I learn from my mathematics books is hardly any use to me when I move around in society where I live. The only […]

Prove withoui calculus: the integral of 1/x is logarithmic

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This is. of course, easy to prove with the fundamental theorem of calculus. But is there a simpler, perhaps geometric, way of proving this?

Linear Homogeneous Recurrence Relations and Inhomogenous Recurrence Relations

I’m having some difficulty understanding ‘Linear Homogeneous Recurrence Relations’ and ‘Inhomogeneous Recurrence Relations’, the notes that we’ve been given in our discrete mathematics class seem to be very sparse in terms of listing each step taken to achieve the answer and this makes it incredibly hard for people like myself who are not of a […]

How to know that irrational numbers never repeat?

How would you respond to a middle school student that says: “How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. And, how do they know that numbers like $\pi$ and $\sqrt2$ are irrational because they can’t check an infinite number digits […]

Prove that the exponential function is differentiable

Imagine that you are writing a book on the foundations of analysis. You have already proved that for each $a > 1$ there is a unique function $f_a(x) = a^x$ satisfying the following: $f_a$ is an isomorphism of ordered groups between $(\mathbb{R},+)$ and $(\mathbb{R}_{+},\cdot)$; $f_a(1) = a$. It follows from the monotonicity and bijectivity of […]

How to Make an Introductory Class in Set Theory and Logic Exciting

I am teaching a “proof techniques” class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first order logic. After that we prove simple math statements via direct proof, […]

Roman Numbers – Conversion to decimal number

I have read that if a smaller number is to the left of a larger number means that the smaller number has to be subtracted from the larger number. Ok I can understand quickly for below Roman Numbers : IX = X – I = 10 – 1 = 9 But I have difficulty in […]

Counterexamples to “Naive Induction”

I was teaching a nine-year-old friend about prime numbers. When I asked him if he thought there were finitely or infinitely many primes, he answered confidently that there must be an infinite number. “How do you know?” I asked. “Because I can keep thinking up larger and larger primes. It’s easy!” By way of proof, […]