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Study Guide: What Is Mathematics?

The Ability to Count

Mathematics is simply a language of patterns, similar to music but harder to hear. 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.

Most humans are born with the ability to count up to about four or so. We call this our natural number sense. Many studies show that while infants have no understanding of our human-invented numbering systems, they certainly can identify changes in quantity, such as when their siblings get more candy.

However, our numerical sense becomes less precise with increasingly larger numbers. For example, we are slower to compute $4 + 5$ than $2 + 3$, and must are utterly lost around $39563 + 7568$ As another example, those who put their faith in some day winning the lottery demonstrate near zero understanding of large numbers (and terrible odds).

Speaking in binary, it's worth noting 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 ancient past, we began to develop tricks to support our sense of numbers. We counted on our fingers and toes, or sorted groups of pebbles and sticks. This is why so many modern number systems use groups of five, 10 or 20. Base-10 (the familiar Decimal System) stems from using both hands to count. Base-20, or the Vigesimal System, is based on using both hands and feet.

As Within So Without

Humans apparently have a strong need to anthropomorphize the world. They invent Gods in their own image, imagine that the stars orbit according to their own personal fate, etc. At some point, ancient humans externalize their number sense as well. In doing so they created humanity’s most important scientific achievement: the universal language of mathematics.

Math is Hard!

Although we have a natural ability to count, numbers can still be difficult to work with. Sure, some of us have the gift, but everyone reaches a point where math become hard. Learning the multiplication tables is difficult because we did not evolve to handle such advanced computations as $17 \times 32 = 544$, and until very recently had absolutely no need to.

Math can get you killed

In earlier times, those who sat around solving math problems often failed to notice their surroundings and ended up covered in Dragon Ketchup©, while those who ignored the math and simply hit the ground running lived on to tell the tale.

Number sense may come naturally, but developing mathematical literacy (the ability to read the language of mathematics) 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.

The Tower of Math

Think of math as a tower. Our human height and vision is finite (limited), so if we want to reach higher and see farther, we need to build a tower. Our natural mathematical abilities are also limited, so we have built a tower of mathematics.

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

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 early Christians were so horrified at the idea that their Lord might have been born on the devilish year zero that the number was removed from the calendar. Thus, the year 1 BC is followed by the year 1 AD, editing the poor Year 0 completely out of existence.

The hope was that this would fool the Devil and delight the 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 illogical mess naturally leads us to the next subject—the Rational Numbers.

Rational Numbers

At this point you might be forgiven for thinking that rational numbers are the opposites of irrational integers. However, as so often happens when humans try to think, you would be sadly mistaken. The name ‘rational’ is based on the idea of the ‘ratio’, i.e., a comparison of two integers, often written as a fraction or a decimal. Even integers are rational numbers, as we shall soon see.

For example, $0.5$ is a quite respectable rational number because it can be written as ratio, $\frac{1}{2}$ or $1 : 2$.

Every integer can also be written as ratio (or fraction). The number $3$ can be written as $\frac{3}{1}, and for clarity probably should be.

That’s a good thing because if $3$ did not mean “three of something”. in this case “three of one”, it wouldn’t mean anything at all. Might as well just call it zero!

Irrational Numbers

Early Christians will be pleased to learn that the opposite of the Rational Numbers is not the Christian Numbers. Instead it is… drum roll… the Irrational Numbers! Now these are the scary ones. And true to form people have killed each other over these numbers for quite sometime. Entire philosophical traditions have fallen due to the discovery of a single irrational number. Just ask your neighborhood Pythagorean, most of whom fled to the far corners of the Earth following their shattering discovery of the irrational number $\sqrt{2}$.

Another irrational number that scared the togas off respectable Pythagoreans is the humble $\pi$ (the ratio of the circumference of a circle to its diameter) or $\frac{Circumference}{Diameter}$. Rumor has is that this unfortunate fact so horrified the good Pythagoreans that they strove to keep it forever hidden, and the punishment for speaking of it was death.

Most unfortunately for the Pythagoreans who–believed the Integers and Rational Numbers were symbols of Gods–$\pi$ can NEVER be written accurately as a ratio of two integers. In fact not only can’t we write $\pi$ as an integer, we can accurately write it at all. We simply don’t know exactly what $\pi$ is! The best we can say is that it’s somewhere between two other numbers, such as less than $4$ and more than $3$.

Here are some approximations for $\pi$:

There’s much more to $\pi$ than meets the eye. Discover more here:

• Wikipedia: https://en.wikipedia.org/wiki/Pi

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

Real Numbers

As humanity began to understand deeper aspects of reality, our mathematical language evolved. This pleases us greatly because if we can reduce something to numbers, we tend to think we understand it.

Rational and irrational numbers are termed the Real Numbers because they can all be found on an infinitely long Number Line, conveniently called… The Real Number Line.

But! There are other numbers that exist far beyond the real number line! If this strains your imagination, you are not alone. These numbers are therefore called, The Imaginary Numbers.

Complex and Imaginary Numbers

A complex number is any number which can be written as $a + bi$, such that $a$ and $b$ are real numbers, and $i^2 = -1$. This number $i$ is called the imaginary unit, and it is defined so that its square equals -1.

If you understand the idea of square numbers, at this point you would be forgiven for exclaiming that no square number (any number multiplied to itself) can ever be negative. You are correct! Consider:

So be it! We needed to invent the complex numbers for our mathematics to deal with certain complex aspects of reality, and so we did. Period. Remember, mathematics is only a language.

In a quadratic equation, we learned that the value under the square root, $b^2-4ac$, is called the discriminant, and that if the discriminant is negative, then the quadratic function has no real roots because we can’t take the square root of a negative number.

But! When using imaginary numbers, this is no longer true! We can find square roots of negative numbers using imaginary numbers, and every quadratic equation will always have exactly two complex roots! That’s a result with huge implications for the increased usefulness of math.

Transcendental Numbers

The story of numbers continues with the Transcendental numbers, but I fear I will lose most readers here. For more on the transcendental numbers see this: https://en.wikipedia.org/wiki/Transcendental_number.

As our understanding of reality expands, we keep discovering/creating new types of numbers, all of which become part of the language of mathematics.

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?

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?