Math

We found 43 articles

Proof by contradiction: how to be so wrong you end up being right
Mathematical proofs are what make math objective: while you could find a few examples that "prove" a mathematical statement, it is often more important to write a rigorous proof that holds true in all cases. Mathematicians have a few methods in their toolkit to tackle different proofs. In this post, we will learn how to write a proof by ...
Solving algebraic equations with variables on both sides
If you are reading this, I can tell you’ve mastered solving simple linear equations. You’ve mastered the art of balance. You know that whatever you do to one side of the equal sign, you must do to the other. You can perform inverse operations until the cows come home, and you are a pro at isolating the variable. I bet you even check your work by ...
How to solve linear algebraic equations
Today, we are going to learn how to solve linear algebraic equations like 3y + 3 = 18 or 5x - 4 = 16. If these equations make you feel a bit queasy, have no fear! I am going to break the process down into five simple steps. 
Calculating flood risk using probability and statistics
If at some point you ever want to buy property near water, a variation of this question will undoubtedly pass through your head: what are the chances that my {insert name of your expensive piece of property close to water} floods?
The Intermediate Value Theorem explained by everyday life
Calculus can be tough stuff. Calc AB was the first AP class I ever took in high school, and though I love the subject now, I certainly didn’t love it when I was first struggling with limits or with the chain rule for derivatives.
How to approach initial value word problems
We’ve all been there: on a homework set or in an exam, you turn to the final page and, to your dismay, it’s a wall of text. The dreaded Word Problem. Some of the words are useful, some of them are meant to distract. Let’s look at a strategy for answering initial value word problems.
Gaining intuition for Taylor series
I’m writing this blog post because when I first came across Taylor series I found that a lot of my previous intuitions for mathematics were suddenly inadequate. It took time for me to build intuition not only for how these things work but why they are important. I hope this post will be illuminating for those just beginning to learn about Taylor ...
Where do Taylor series come from and why do we learn about them?
Taylor series can often seem a bit mysterious the first time that we learn about them. The formula for the Taylor series of a function f(x) around a point x=a is given by
What is implicit differentiation and how does it work?
One topic that seemed a bit mysterious and magic to me when I first learned calculus was implicit differentiation. In this post, we’ll start by reviewing some examples of implicit differentiation and then discuss why implicit differentiation works.
But what is “dx” really? Calculus terms explained
The symbol “dx” comes up everywhere in calculus. For example:
The Idea of Taylor Approximation
If you type √5 into your calculator, it’ll output something like 2.2360679775. But how did your calculator find that answer? Is there any way you could have found it by hand?
How to survive a proof-based math class
Probably the most common challenge that I see my students struggle with is understanding and writing out mathematical proofs. Although most higher-level college math and computer science courses rely heavily on proofs, there aren’t many courses that really prepare students before they’re thrown off the deep end. I wanted to discuss some tips and ...
An Application of Calculus: Finding Optimal Road Networks
Suppose that we have many towns spread across the country and we are trying to connect them with a network of roads. If we would like to do so by laying as little road as possible, how do we do it? In this blog post, we will use Calculus to tackle a special case of this optimization problem.
Which is bigger?: Set cardinality, injective functions, and bijections
Comparing finite set sizes, or cardinalities, is one of the first things we learn how to do in math. From a young age, we can answer questions like “Do you see more dogs or cats?” Your reasoning might sound like this: There are four dogs and two cats, and four is more than two, so there are more dogs than cats. In other words, the set of dogs is ...
What is Mathematical Induction (and how do I use it?)
Mathematical induction is a common and very powerful proof technique. At its core, it’s an appeal to an intuitive notion that Induction proofs often pop up in computer science to proof that an algorithm works as intended (correctness) and that is runs in a particular amount of time (complexity). In this tutorial we’ll break down a classic ...
An Insider's Tip to Prepping for the SAT Math Section: Plug in Numbers
Preparing for both mathematics sections on the SAT can be a bit intimidating. You can’t expect yourself to know every topic that might come up, and the time limit adds to the stress. Much more efficient than trying to learn everything you might come across is to start with what you have already learned in high school and use examples to apply it ...
The Key to Mastering Mathematics? Quit Memorizing.
There are many misconceptions when it comes to the subject of mathematics.  One of the most common myths I encounter is related to the way one approaches learning math. 
How To Multiply Matrices Quickly and Correctly in Six Easy Steps
Step 0 – Make Sure the product makes sense! Say we’re given two matrices A and B, where
How to Study for a Math Test: 4 Essential  Tips
It’s a Tuesday night, and you’ve just remembered that you have a huge algebra 2 test tomorrow. Panic starts to sweep over you as you think back on all you’ve learned this semester. How will you ever succeed? You call up your friend to ask how you can study, and he calmly tells you he has just spent about an hour reviewing the main concepts from ...
How Majoring In Math Will Change The Way You See The World
There is a humorous, misguided stigma associated with math. Just the mere utterance of the word “math” conjures up, for many, the image of intimidating, arcane equations strewn about blackboards and calculators gone on the fritz. For many, math was a painful experience in grade school (and beyond) -- myself included! In fact, I did not become ...
Tips for Getting a Perfect Score on a Standardized Math Test
SAT, ACT, SSAT, ISEE, GRE. What do these acronyms all have in common? Well, they’re all standardized tests, but more importantly, they all have multiple-choice math test sections. Despite whether or not they’re accurate indicators of student performance in the classroom, lab, or office, they are all essential for entry into some educational career ...
Solving the “I’m not good at math” problem
You’ve heard it before. Or you’ve said it. I’m not good at math. I hear it from seventh graders struggling with fractions, high school students preparing to take the SAT, friends at a restaurant when splitting a check, and even from parents assuring me that their child’s own difficulties are in fact genetic. And while I’ve heard it countless ...
How to Understand Matrix Factorization
Don’t let the word “matrix” scare you off! Even if your only experience with them is from the movie The Matrix, you know enough to learn about how they can be used to recommend you movies you might like, including, if you haven’t seen it already, The Matrix.  Today, we'll go over matrix factorization by taking a look at Netflix!
How to sketch any graph by eye
Equations in math are useful but they’re also kind of inefficient – for each x value, you have to do a separate calculation to figure out what y is. Graphs take that equation and turn it into a visual, something you can look at and immediately see what happens at different values of x, how the function changes, and more!
Four Mathematicians You Should Know
Math has changed a lot over the years. When most people think of math, they likely think of someone sitting quietly at a desk with a book or some paper. It’s an unmoving image. When we think of people who are good at math, we conjure up people who blaze through problems quickly and alone. They follow the rules in math and in life.
What is Spectral Geometry?
All shapes, from strings to bridges to carrots, resonate You’re having breakfast in the kitchen when you start to hear a series of slow and repetitive thuds – someone is coming down the stairs. Without having to ask or look up, you can instinctively guess who it is. That’s because the cadence and volume of one’s footsteps is unique from person to ...
What is a Limit, Anyways? Mathematical Concepts Explained
Let’s talk about a concept that can be confusing when you’re first studying calculus: limits. When you’re first introduced to limits, you’ll often hear your professor say things like, “What is the limit of f(x) =  as x approaches 5?” When worded like that, limits don’t sound very natural or intuitive – but in today’s post, I’m going to convince ...
How to Help Your Child with Math Homework: 5 Easy Questions You Can Ask
If you’re the parent of a teenager, chances are good that a few years have passed since you had to graph a polynomial or find a derivative. Since high school math covers topics that people working outside of STEM don’t come across very often, many parents don’t feel like they can give much help to their teenage children with their math homework. ...
Revolving Curves to Make Solids
Have you ever wondered where the formulas for volumes that you studied way back in geometry come from? It’s not too surprising that the volume of a cube is , but why is the volume of a cone ?
Data Science and Intuitive Mathematics
Sitting at the cross-roads of mathematics, statistics, and computer science, the emerging field of data science (ranked by many as the top career in the US) seems daunting to those still developing strong technical skills. At the same time, a host of dynamic and highly-efficient libraries give coders the power to treat complex areas like machine ...
Why logarithms are actually useful: Simplifying Arrhenius temperature dependence using log tricks
Learning about logarithms is one of those times in math class where you wonder if this will ever be useful in any way. I see lots of students struggle with topics like logs, since they can seem abstract and they aren’t obviously useful. But I’m here to explain why they are actually incredibly important and describe so much of the world we live in! ...
Converting Polar to Cartesian Equations in Five Easy Steps
If polar equations have you second-guessing your future as a nuclear physicist, fret not!  Almost every pre-calculus student I have tutored has struggled here, and it isn’t surprising at all.  Remember the first time you saw an equation and were introduced to these strange x and y variables?  It may seem like second nature now, but you were ...
Math Tips: Triangles
Triangles can be very different! Triangles are the most basic 2D shape that you can create and they can be found everywhere. So let’s look at some tricks and facts about triangles. We’ll start by looking at the triangle below (not drawn to scale).
The ambiguous case of the law of sines explained
Trigonometry should be simple—you’re just using the given information to solve for only one answer, right? Well, with the Law of Sines, sometimes there is more than one right answer. This situation is also known as the Ambiguous Case.  Before we dive into the Ambiguous Case, let’s review the Law of Sines and Congruence.
The Math Tutor: Simplifying Probability
Dungeons and Dragons is complicated. Mastering probability doesn’t have to be. Probability is one of those topics that haunt children from grade school days of the determining the likelihood of picking out red marbles from a box. Even my most advanced math tutoring students in Cambridge sometimes feel bamboozled by it. Why? Because probability and ...
The Math Tutor: How Geometry Can Help You to Improve at Proving
If you’re taking a Geometry class, chances are that you’ve spent the last couple of months learning about proofs (and if you haven’t gotten there yet, you will soon). Proofs are something students either love or hate, but mostly they love to hate them. Why, they often ask me, are we learning how to do this? How is this relevant to anything? I’m ...
The Math Tutor: 3 Anecdotes from Great Mathematicians’ Lives
It’s a shame that so many people can go through college as math majors and minors without ever learning the history of mathematics. Who were Euler and Gauss? Newton and Leibnitz? Euclid? We all know their theorems and mathematical contributions, but rarely do most of us think of the people —with their messy lives, quirks, and stories— behind these ...
Math Tutor: 4 Easy Steps to Volume Rotation Integration
Merry Christmas Eve! Why yes, I do believe volume rotation integration is as easy as apple pie!  Every once in a while something you learn in class really just knocks you off your feet. Those of you who have taken AP physics will know that position is the integral of velocity. About a month ago, on a fairly dreary Boston morning, I was sitting in ...
Math Tutor: Tackling Words Problems
What is the first thing that pops in your mind if you think about math? If you are anything like most people I know, you probably think about long equations and numbers. After all, that’s what math is, isn’t it?