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Study Guide: What Is Mathematics?

Mathematics is simply a language of patterns. We identify patterns in the world around us to survive the challenges of life. To do this well, we need a reliable sense of numbers.

For example, if some day we find ourselves wandering down an ancient jungle path only to run into one angry bear on the right and five angry bears on the left—with all else being equal—every self-respecting humanoid knows to sit down, put on their running shoes, and sprint like a bat outa hell to the rear. OK, bad example, but that’s the point. You already know how to make the example better.

Number sense plays a vital role in the way all animals navigate their environment—an environment filled with objects that are very mobile, often dangerous, and sometimes great with Ketchup(r).

Let’s say you walk outside your home and see two dogs playing. Even if you didn’t know the word “two” or what the corresponding number looks like, you have a good sense of how a two-dog encounter compares with a three-, four- or zero-dog situation.

The Ability to Count

From birth, we have a natural ability to count. We call this number sense. Many studies show that while infants have no understanding of our human-made numbering systems, they can identify changes in quantity.

However, our numerical sense becomes less precise with increasingly larger numbers. For example, we are slower to compute $4 + 5$ than $2 + 3$. Those who put their faith in someday winning the lottery show that some people have almost zero understanding of large number (or terrible odds).

Speaking in binary, it's worth remembering that there are 10 kinds of people in the world. Them that can count, and them that can't.

The Invention of Numbers

At some point in our very ancient past, we began to develop tricks to support our number sense. We started counting on our fingers and toes or by sorting groups of pebbles and sticks. This is why so many modern number systems use groups of five, 10 or 20. Base-10, or the decimal system, stems from using both hands to count. Base-20, or the vigesimal system, is based on the use of all the fingers and toes.

So ancient humans learned to externalize their number sense and, in doing so, they created humanity’s most important scientific achievement: the universal language of mathematics.

Although we have a natural ability to count, numbers can still be difficult to work with. Sure, some of us have a gift for math, but every one reaches a point where math become hard. Learning the multiplication tables is difficult because we did not evolve to handle such advanced computations as $17 \times 32 = 544$.

In earlier times, those who sat around trying to solve such problems often ended up covered in Dragon Ketchup(r), while those who ignored the math and simply hit the ground running for their lives lived on to tell the tale. (If this were not so, we would not know about dragons.)

Number sense may come naturally, but developing mathematical literacy (the ability to read math) takes time and effort. Meanwhile, things just keep getting harder! Humanity’s use of mathematics has steadily grown over the ages. Like science itself, math isn’t one person’s invention. It is a steady accumulation of knowledge throughout human history.

The Tower of Math

Think of math as a tower. Our human height is finite (it has limits), so if we want to reach higher into the air to see farther across the landscape, we need to build something. Our natural mathematical abilities are also limited, so we have built a great tower of number systems to climb toward the stars.

Types of Numbers

Numbers are words and symbols representing quantities. To break down the basic structure of the Tower of Math, let’s first look at the raw materials. Here are the basic types of numbers:

Integers

You probably know integers as whole numbers. But the plot thickens fast. They come in both positive and negative forms, each being the mirror opposite of the other. Integers include the basic counting numbers ($1, 2, 3…$), negative numbers ($-1, -2, -3…$) and everyone’s favorite, the almighty zero ($0$), which stands alone as neither positive nor negative, and without equal nor opposite.

Zero is so strange that people killed each other over it. Note that the Gregorian Calendar still in use today does not include the year zero. This is because the early Christians were so horrified by that devilish number that they choose to name the year following 1 BC as 1 AD, editing the poor Year Zero right out of existence.

The hope was that this would fool the Devil and delight God. Maybe they were successful, and God yet smiles upon the face of the Earth, but it seems that this sinful world has been off by exactly one year ever since. This leads us to the rational numbers.

Rational Numbers

At this point you might be forgiven for thinking that rational numbers are the rational cousins of those God-fearing integers. However, you would be sadly deluded. Rational numbers are simply ratios of two integers. Rational numbers can be written as fractions. Nothing more; nothing less.

For example, $0.5$ is quite respectable rational number because we can be written as $\frac{1}{2}$. Note that every integer can be written as fraction. For example the number $3$ can be written as $\frac{3}{1}. That’s a really good thing because if $3$ did not mean “three ones” it really wouldn’t mean much of anything at all. Might as well just be a zero.

Irrational numbers

Irrational numbers are the scary ones. And true to form people have killed each other over these numbers. Entire philosophical traditions have fallen due to one irrational number. Just ask your neighborhood Phythagorean. Oh! I forgot! They fled to the far corners of the Earth following the shattering discovery of the $\sqrt{2}$.

$\pi$ (the ratio of the circumference of a circle to its diameter) is a classic example. $\pi$ can not be written accurately as a ratio of two integers. In fact not only can’t we write $\pi$, we don’t even know what $\pi$ is! The best we can say that it’s somewhere between two other numbers, such as less that $4$ and a little more than $3$.

Here are some approximations for $\pi$:

$$\pi = \dfrac{355}{113} \text{ } \dfrac{22}{7} = 3.142857142857…$$

There’s a lot more to $\pi$ than meets the eye. Learn more here:

Question: What do you get if you divide the circumference of a jack-o-lantern by its diameter?

Pumpkin $\pi$
Real Numbers and Imaginary Numbers

As the language of mathematics grows, we develop more ways to understand numbers. Rational and irrational numbers both fall under the categories of real numbers and complex numbers. And yes, there are also imaginary numbers that exist outside the real number line. There are even transcendental numbers. We keep discovering new types of numbers, and they all become a part of the language of math.

The Branches of Mathematics

Arithmetic

This is the oldest and most basic form of mathematics. Arithmetic chiefly concerns the addition, subtraction, multiplication and division of real numbers that aren’t negative.

Algebra

The next level of mathematics, algebra, is arithmetic using unknown quantities. We represent the unknown parts with symbols, such as x and y.

Geometry

Geometry was original developed to help early humans find their way around. It deals with the measurement and properties of direction, distance and size. Geometry requires an understanding of points, lines, angles, areas and volume.

Trigonometry

Trigonometry is used to measure triangles and the relationships between their sides and angles. While the historical origins of arithmetic, algebra and geometry are lost in the fog of ancient history, we know that trigonometry was developed the second century by the great astronomer Hipparchus of Nicaea.

Calculus

Independently developed by both Isaac Newton and Gottfried Leibniz in the 17th century, Calculus deals with the calculation of instantaneous rates of change. It focuses on finding the kinds of answers that may be very close to zero or infinity.

Invented or discovered?

The tower of mathematics has enabled human culture to rise and flourish, to speak across time and space, and to understand both the inner mysteries of life to the outer mysteries of our world. But did we truly build this tower out of our own ingenuity?

Riddle: How do mathematics teachers scold students?

“If I’ve told you $n$ times, I’ve told you $n+1$ times…”

Questions

  1. What is your natural mathematical ability? For example, how many things can you instantly sense without needing to think or count?
  2. What is the difference between arithmetic and algebra?
  3. Was mathematics invented or discovered?
  4. How do we know mathematics is true or accurate?
  5. Is mathematics true, everywhere, always?
  6. Do you think there might be a place or time where our mathematics is wrong? What would that place be like? How would it be different from our world?
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