Introduction to Mathematics


I’m Professor Dave, and I wanna teach you
about mathematics. Not everyone loves math. Many students wonder why it’s even necessary
to learn in the first place. Well, it may not always be obvious, but math
really is everywhere. Math is how our phones work, how airplanes
fly, how social media knows who you’re going to tag in a photo before you do. Math can tell us how disease spreads, who
is most likely to win an election, or the odds of winning at casino games. Math also goes far beyond our everyday experience. It is the language of the universe itself,
describing how all objects move, from tiny atoms to entire galaxies. Math governs everything from science to economics,
so it is not an overstatement when we say that math is everywhere. But in another more tangible sense, math is
also nowhere, in that it isn’t a concrete physical object, like you, or your shoe, or
your desk. It is inherent in reality. Objects in the night sky obey mathematical
laws of motion that exist whether we write them down or not, which means that mathematical
truths have their own existence, independent of humans and our symbols. The fact that one plus one equals two does
not rely on two people each putting an apple into a basket and then agreeing that the resulting
is two apples. It’s just true. The equation that represents this action was
true before any humans were even born, and it will be true long after they all die. So where did the math we see in math class
come from? Did some caveman stumble across a Calculus
textbook buried under a rock? Is this knowledge bestowed upon us from the
heavens, like a religious text? For those of us who have difficulty with math,
it definitely seems that way sometimes. But the real story is much more interesting. Math comes from us. We make it. And this does not strictly refer to the past. Every day, thousands of mathematicians around
the world are busy creating more math. So how do we create math? To answer this, we need to know what math
really is. In short, math is the study of questions that
have definite answers. That’s it. This may seem incredibly broad, and it is,
but this is what sets math apart from other areas of inquiry. Some questions don’t have definite answers. What is the best form of government? Who deserves the Oscar for best picture? What’s better, pizza or cheeseburgers? The answers to these questions are subjective;
they vary from person to person, and no matter what anyone says, no one can give a definite
answer, because there simply isn’t one. Even with science, if we do an experiment
over and over and over, and our equations always make highly accurate predictions about
what will happen, we are very likely to be right about a certain phenomenon. But we can never be one hundred percent certain
that we won’t get a different result in the future. Science can be consistent far beyond reasonable
doubt, but it never claims complete certainty. However, if I draw a circle and try to determine
the angle between a radius and a tangent line, I will either get ninety degrees, or I will
be wrong. There are no two ways about it. Every mathematical question like this one
has a completely inarguable answer. Are there infinitely many prime numbers? Yes, there definitely are. Is there a fraction that, when multiplied
by itself, gives the number two? No, there definitely isn’t, and we will
learn about these concepts in this course. The job of the mathematician is to ask these
kinds of questions and then come up with the tools that are needed to answer them. The answers themselves are fixed and completely
out of our control. They are part of the fabric of the reality
in which we live. Often times these answers are very surprising. If we ask the question: “Is there more than
one kind of infinity?”, the answer is yes. In fact, there are infinitely many different
kinds of infinities. Humans did not decide that this was true,
we simply decided to ask the question. And that is what we’re going to do in this
course. We’re going to ask questions, and then build
the tools we need to answer them. The tools that we will build can be applied
in virtually every aspect of life, from the biological, to the technological, to the sociological,
and even more astoundingly, they sometimes show us the deepest of truths that have been
lying in wait for us to uncover. These are the ones that tell us about the
nature of reality in a way that philosophers can only dream. So if you want to build up your understanding
of math so that you can learn all kinds of physics and chemistry and astronomy, you’ve
come to the right place. If you’re a math student at any level, and
you need help understanding what your teacher is talking about in class, you’re in the
right place, too. If you’re just someone who wants to figure
out how much tip to leave at dinner, stick around, all are welcome. In this course, we will begin with the most
elementary math there is, the basic arithmetic that was developed at the dawn of civilization,
and work our way through everything one would learn in high school, and most of an undergraduate
mathematics education. Watch all the way to the end, or bail when
you feel like you’re in over your head. Either way, I bet you’ll get farther than
you thought you could. So grab a pencil, and let’s learn some math.