There are infinitely many numbers, and infinitely many ways to combine and manipulate those numbers.
Mathematicians often represent numbers in a line. Pick a point on the line, and this represents a number.
At the end of the day, though, almost all of the numbers that we use are based on a handful of extremely important numbers that sit at the foundation of all of math.
What follows are the eight numbers you actually need to build the number line, and to do just about anything quantitative.
In The Beginning, There Was Zero.
Zero represents the absence of things. Zero is also an essential element of our number system. We use zero as a placeholder when writing numbers with more than one digit — zero lets me know the difference between having 2 dollars and 20 dollars.
Zero as a number on its own is also extremely important in math. Zero is the “additive identity” — any time I add a number to zero, I get that number back: 3 + 0 = 3. This property of zero is a central aspect of arithmetic and algebra. Zero sits in the middle of the number line, separating the positive numbers from the negative numbers, and is thus the starting point for building our number system.
We Can’t Get Very Far Just Having Zero, So We Turn To One.
As zero was the additive identity, one is the multiplicative identity — take any number and multiply it by one, and you get that number back. 5 x 1 is just 5.
Just using one, we can start to build up the number line. In particular, we can use one to get the natural numbers: 0, 1, 2, 3, 4, 5, and so on. We keep adding one to itself to get these other numbers: 2 is 1 + 1, 3 is 1 + 2, 4 is 1 + 3, and we keep going, right on out to infinity.
The natural numbers are our most basic numbers. We use them to count things. We can also do arithmetic with the natural numbers: if I add or multiply together any two natural numbers, I get another natural number. I can also sometimes, but not always, subtract two natural numbers, or divide one natural number by another: 10 – 6 = 4, and 12 ÷ 4 = 3. Just using zero and one, and our basic arithmetic operations, we can already do a good amount of math just using the natural numbers.
Natural Numbers Are Pretty Great, But They Are Also Quite Limited.
To start with, it is not always possible to subtract two naturals and get another natural. If all I have to work with are these counting numbers, I have no idea how to parse a statement like 3 – 8.
One of the wonderful things about math is that, when we are confronted with a limitation like this, we can just expand the system we are working with to remove the limitation. To allow for subtraction, we add -1 to our growing number line. -1 brings with it all the other negative whole numbers, since multiplying a positive number by -1 gives the negative version of that number: -3 is just -1 x 3. By bringing in negative numbers, we have solved our subtraction problem. 3 – 8 is just -5. Putting together the positive numbers, zero, and our new negative numbers, we get the integers, and we can always subtract two integers from each other and get an integer as the result. The integers provide the anchor points for the number line.
The negative numbers are useful in representing deficits — if I owe the bank $ 500, I can think of my bank balance as being -500. We also use negative numbers when we have some scaled quantity where values below zero are possible, such as temperature. In the frozen wasteland of my hometown of Buffalo, we would get a few winter days each year down in the -20° range.
Integers Are Suited To Describing Whole Numbers, But We Need To Talk About Fractions Of Things.
Also, the integers are still arithmetically incomplete — while we can always add, subtract, or multiply two integers and get another integer, we cannot always get an integer by dividing two integers. 8 ÷ 5 makes no sense if all we have are whole numbers.
To deal with this, we add 1/10, or 0.1, to our number line. With 0.1, and the powers of 0.1 — 0.01, 0.001, 0.0001, and so on, we can now represent fractions and decimals. 8 ÷ 5 is now just 1.6. Dividing any two integers (except for dividing by zero) gets us a decimal number that either terminates, like 1.6, or has a repeating digit, or pattern of digits: 1 ÷ 3 = 0.3333…, with the 3′s going out to infinity. These types of decimals are the rational numbers, since we can form them by taking fractions, or ratios, of two integers. The rational numbers are arithmetically closed – I can take any two rationals and add, subtract, multiply, or divide them, and get back another rational number.
The rational numbers allow us to represent quantities between integers, or fractional quantities. If three friends and I are sharing a cake and splitting it up evenly, we each get 1/4, or 0.25, or 25% of the cake. The rationals help us start filling in the spaces between the integers on the number line.
The Square Root of 2
Rational Numbers Open The Door For Irrational Numbers.
The square root of a number is a second number that, when squared, or multiplied by itself, gives us the original number. So, the square root of 9 is 3, since 32 = 3 x 3 = 9. We can find the square root of any positive number, but with only a few exceptions, these square roots get messy.
The square root of 2 is one such messy number. It is an irrational number — its decimal expansion never terminates or settles into a repeating pattern. The square root of 2 starts out with the digits 1.41421356237…, and then just keeps going in weird and random-looking directions. It turns out that the square roots of most rational numbers are irrational — the exceptions, like 9, are called perfect squares. Square roots are important in algebra, as they figure into the solutions of many equations. For example, the square root of 2 is a solution to the equation x2 = 2.
By putting the rational and irrational numbers together, we complete our number line. The full range of rational and irrational numbers are called the real numbers, and these are the numbers most commonly used in all manner of calculations. Now that we’ve completed our number line, we can look at a couple really important irrational numbers.
Let’s Add Another Dimension.
π, the ratio of the circumference of any circle to the circle’s diameter, is maybe the most important number used in geometry. π shows up in basically any formula involving circles or spheres — the area of a circle with radius r is πr2, and the volume of a sphere with radius r is (4/3)πr3.
π also features prominently in trigonometry. 2π is the period of the basic trigonometric functions sine and cosine. This means that the functions repeat themselves every 2π units. These functions, and thus π, are key to working with any periodic or repeating process, particularly in describing things like sound waves.
Like the square root of 2, π is irrational – its decimal expansion never terminates or repeats. The first few digits of π are pretty familiar — 3.14159… Mathematicians using really big computers have found the first 10 trillion or so digits of π, though for most day to day applications, we only need those first few digits to get sufficiently precise results.
Euler’s Number (e)
Introducing The Foundations Of Computing Compound Interest.
Euler’s number, e, is foundational to working with exponential functions. Exponential functions represent processes that double or halve themselves in a fixed period of time. If I start with two rabbits, after a month I will have four rabbits, after two months I will have eight rabbits, and after three months I will have 16 rabbits. In general, after n months, I will have 2n+1 rabbits, or 2 multiplied by itself n+1 times.
e is an irrational number, approximately 2.71828…, but like all other irrational numbers, the decimal expansion goes on forever with no repeating pattern. ex is the natural exponential function, the baseline for any other exponential function. The reason ex is special is a little complicated. For those of you who have seen calculus, you may know that the derivative of ex is ex. This means that, for any particular value of x that we plug into ex, the rate at which the function is growing at that point is the value of the function — for x = 2, the function ex is growing at a rate of e2. This property is basically unique among functions, making ex very nice to work with mathematically.
ex is useful in working with most exponential processes. One of the most common applications is finding compound interest that is being compounded continuously. With a starting principal of P, and an annual interest rate r, the value of an investment A(t) after t years is given by the formula A = Pert.
The Square Root of -1: i
And Now, Imaginary Numbers.
We mentioned earlier that we can take the square root of any positive number, so now we see what happens with negative numbers. Negative numbers do not have square roots in the real numbers. Multiplying two negative numbers together gives you a positive number, so squaring any real number results in a positive number, so there’s no way to multiply a real number by itself to get a negative number. But as we saw earlier, when we are confronted with an apparent limitation like this in a number system, we can just expand the number system to remove the limitation.
And so, confronted with the limitation that we do not have a square root for -1, we simply ask ourselves what would happen if we did. We define i, the imaginary unit, to be that square root, and by throwing in all the other “numbers” we need to make sure that addition, subtraction, multiplication, and division still make sense, we extend the real numbers to form the complex numbers.
The complex numbers have many amazing properties and applications. Just as we were able to represent the real numbers as a line, we can represent the complex numbers on a plane, with the horizontal axis representing the real part of the number, and the vertical axis an imaginary component, representing the square root of some negative number. Any polynomial equation has at least one solution in the complex numbers, a result so important that mathematicians call it the fundamental theorem of algebra. The geometry of the complex plane results in some surprising and elegant results, and has many applications in the physics of electricity and in electrical engineering.