What are real numbers? Basically, they are any number that you could use to measure a distance, including all positives, negatives, and zero—both whole and fractional. Practically any number you can think of, 8, -5, 3/4, -9.21523, √2, π (“pi”), are all real numbers.
So what in the world does a non-real number look like? Well, believe it or not, there are actually imaginary numbers, and we’ll talk about all that below.
And now let’s explore the top five Things You Should Know About Real Numbers in Math!
1. The Real Numbers Include Integers, Rational, and Irrational Numbers
Ok, first let’s talk about how mathematicians classify numbers.
Natural Numbers — We’re Counting on You!
The usual counting numbers, 1, 2, 3, 4, 5, … are called the natural numbers.
Natural numbers are, well, natural! They’re the first concepts in math that toddlers learn. They helped our distant ancestors keep track of the number of sheep and goats they owned, and facilitated trade. Your street number is (probably) a natural number, as is your phone number. They’re all around us!!!Image By John Carnemolla
As simple as the natural numbers are, they have a surprising property: they never end! There is no “largest” natural number. In the language of mathematics, we say that the set of natural numbers (and therefore, the set of real numbers) is infinite.
Into the Abyss — Negatives and Zero
When you start counting more abstract things like money, you may find that the natural numbers are not quite sufficient.
For example, suppose that you have a bank account with $100 in it. You can think of this deposit as a record that the bank owes you $100.
Every day, you use your bank card to buy lunch for $10. Let’s keep track of the money in your account day by day.
100, 90, 80, 70, 60, …
You get the drift right? So far, we can represent the amount of money using natural numbers.
However, what happens after ten days of buying lunch? Your stomach may be full, but your bank account is empty! All of the 100 dollars is gone, leaving nothing in your bank account. We represent that “nothing” as the number 0.
Then what happens if you decide to swipe your card the next day at the buffet? With no money in the bank, you would then owe the bank $10. And instead of seeing a nice natural number of dollars on your balance sheet, you’ll see either -$10 or ($10), depending on how the bank chooses to record it. Congratulations: your balance is now negative.Image by schuldnerhilfe
So, the bottom line is that we needed special numbers like 0 and negative numbers to help measure certain kinds of quantities like profit and loss.
The natural numbers, their negatives, and zero constitute the set of integers.
Reading Between the Lines — Rational and Irrational Numbers
Integers are pretty good at keeping track of individual things like sheep or dollars. But they’re not so good with lengths and distances.
If you’ve ever built anything from wood, you know that the cuts have to be very precise. What if you want the table legs to be a little bit more than 2 feet? What is a “little bit”? Without a way to accurately measure fractions of a foot, none of the four legs of the table would match each other in length!
Carpenters have long ago solved that problem by breaking up their measurements into smaller and smaller fractions. Inches are a fractional part of a foot. Then inches break down even further to eighths, sixteenths, etc.
Similarly, there are fractional numbers that exist between any two integers. In fact the number line is littered with them! Between any two fractions exists another fraction.
The set of all integers and fractional numbers between them comprise the set of rational numbers.
But we’re not done yet! Even rational numbers don’t cover everything.
For example, consider a right triangle whose base and height are both one unit long. The Pythagorean Theorem tells us that the hypotenuse (long side) must be √2 units long.
It can be shown (proven) that √2 is not equal to any fraction of whole numbers. In other words, this number is not part of the set of rational numbers.
Any number on the number line that is not rational is called irrational.
All Together Now!
So now we can answer the question, what are real numbers?
- A real number is any number that measures a length, including their negatives and zero (0).
- The set of real numbers is divided into two types: rational and irrational.
2. The Number Line Contains all the Real Numbers (and Nothing Else)
If we define real numbers as numbers that measure length, then we naturally get a geometric interpretation of the set of reals.
Every real number can be represented by a point on a continuous number line. Conversely, every point on the number line corresponds to a particular real number.
In fact, you can think of the entire number line (or real number line) as a graphical representation of absolutely every real number in existence!
3. Every Real Number Has a Decimal Representation
Every real number can be represented by a numerical string of digits, possibly continuing forever. This is called the decimal representation of the number.
Integers, like 3, -2, 0, 42, and -5,658,142,235 are already in their decimal representation. However, sometimes an integer masquerades in other forms. For instance, 106 is an integer — but you have to expand it to get its decimal form: 106 = 1,000,000.
Rational numbers also have decimal representations. But you have to use a decimal point if the number is not whole.
For example, the number 1/20 is a real number that happens to be rational. As a decimal, 1/20 = 0.05. We call that a terminating decimal.
The real number -4/3 (also rational) has a repeating decimal representation: -1.33333… = -1.3.
Reals often have decimal representations that don’t terminate nor fall into any repeating pattern. For example, 3.1415926… is a non-terminating decimal representation for the real number π.
4. Real Numbers Can do Arithmetic
You can do a lot of cool things with real numbers.
Given any two real numbers, x and y, their sum x + y is also a real number. Their difference x – y, and their product x · y are also real numbers.
The quotient, x / y of two reals is a real number as long as y ≠ 0. On the other hand, x / 0 is not real — in fact, division by 0 is undefined in mathematics.
You can also take powers of real numbers. However, you have to be very careful with roots.
- Any root of a positive real number or zero results in a real number.
- An odd root of a negative real number produces a negative real number.
- An even root (which includes the square root) of a negative real number does not evaluate to a real number. Instead, even roots of negative numbers are imaginary numbers.
So, is real, and equals -2.
But, √-9 is not a real number. Instead, √-9 = 3i, which is an imaginary number.
This leads us to our final important fact.
5. There Are Numbers That Are Not Real
You have to be willing to think outside the box, or at least off the number line, if you want to find numbers that aren’t real.
I mentioned above that even roots of negative reals are called imaginary numbers. In particular, the most famous imaginary number is i, which is defined as the square root of -1. There is no point on the number line corresponding to i. It just sorta lives in its own universe.(Technically, imaginary numbers are on their own number line that exists at right angles to the real number line!)
Every other imaginary number is just a multiple of i. So, 3i, -6.52i, and πi, are all imaginary. Be careful though: 0i is real, because 0i = 0.
A complex number is any sum of a real number and imaginary number. That is, anything of the form a + bi, where a and b themselves are real.
Complex numbers are not real, unless b = 0.
You tend to see complex numbers when solving some quadratic equations. But they crop up in other places in mathematics as well. In fact, they play a significant role in physics, in the areas of electromagnetism, particle physics, and the list goes on.
And so, here’s your Top Five Real Number Facts one more time.
- The real numbers include integers, rational, and irrational numbers.
- The number line contains all the real numbers and nothing else.
- Every real number has a decimal representation.
- Real numbers can do arithmetic.
- There are Numbers that are Not Real (imaginary, complex).