What are Prime Numbers?
A Prime Number is a whole number greater than one, whose only factors are itself and one.
That might sound more complicated than it really is. The thing is, math people like to be very precise with their language; they like to say things in ways that are exactly right. Sometimes that results in simple things seeming more confusing than necessary.
We’ve just honored the math tradition with a precise definition, so next we’ll give a simple explanation. By the end of this page, you’ll have a clear picture of what prime numbers are and also why. Don’t worry — soon it will all make sense.
Before we go on we want to be sure that everyone is clear on what a factor is: factors are numbers that are multiplied by each other to give a product.
Perhaps the simplest way to understand what a prime number is, is to understand what it isn’t.
What are Composite Numbers?
A composite number is the product of two whole numbers greater than one.
That’s worth reading twice, because it’s a precise definition (again). In normal life, if you say something that’s pretty clear, and then a wise guy points out that you’re wrong due to a technicality, you just ignore the wise guy. But, in mathematics, people try to make sure that they’re never wrong, even due to technicalities. If we weren’t worried about details like zero, one, fractions, and so forth, we could have just said: “When you multiply two numbers you get a composite number.” But, being that it’s math, we’re careful about pesky details, even at the expense of being a bit harder to understand.
(The main idea is going to be that composite numbers have factors, and prime numbers don’t. But, in order to truly be correct about it, we have to get into the details, and make sure to dot every i and cross every t.)
By the way, we created a math game to help students become good at breaking composite numbers into their prime factors, because it’s such an important skill. If you are interested in using a game to learn and practice, you can find it on the App Store.
Times Tables Show Composite Numbers.
Here’s a Times Table. We made it only using whole numbers greater than one. The factors are the numbers with a white background on the top and left. The table shows their products with each other in the turquoise colored middle of the table. All those numbers are composite numbers, because they are each products of two whole numbers greater than one.
The table shows a whole bunch of composite numbers up to 100, but there’s a problem.
Does The Table Show All The Composite Numbers up to 100?
No — unfortunately, it doesn’t. For example, 22 is a composite number (it’s 2 x 11), but it’s not in the table because we only multiplied numbers up to 10. However, we know that we do have all the composite numbers up to 10, because when you multiply numbers greater than 1 you always get numbers bigger than you started with. In fact, we know that we’ve got all the composite numbers up through 20, because we’ve tried every number up through 10, and the smallest number they could be multiplied by was 2. Using this logic, if we wanted to find all the composite numbers up to 100, we could make a 50 x 50 times table. However, that would be too much work. Way too much work!! Instead we can use an extremely clever technique called the Sieve of Eratosthenes.
The Sieve of Eratosthenes
We strongly recommend that you make a Sieve of Eratosthenes yourself. It won’t take long, and you’ll learn a lot. Please? By going through the exercise, you’ll really get a deeper feeling for what prime numbers really are and how they work. You’re likely to even start asking some of the same questions that have been catching the imaginations of mathematicians for thousands of years.
When you made your Sieve of Eratosthenes, did you notice that by finding the composite numbers, you also found the prime numbers?
Let’s answer some common questions at this point.
Is 0 a prime number? Is 1 a prime number?
No, neither zero nor one are prime numbers. The simple, but probably not helpful, reason why is that the definition of prime numbers is carefully worded to specifically exclude zero and one. But why would mathematicians care about wording it that way in the first place? They want prime numbers to work like multiplicative building blocks. Zero and one don’t work as building blocks. This way of thinking might make more sense after considering the next question:
Is two really a prime number?
Yes. Although it is even, its only factors are itself and one. If you think of prime numbers as building blocks, two is the building block used to make all other even numbers even. Thinking about it this way, you can see why mathematicians wouldn’t want ‘1’ to be considered a prime number even though its only factors are itself and one. ‘1’ doesn’t work as a building block because you could multiply it into a number a million times and still get the same number. That’s not how building blocks are supposed to work. For example, if you build a composite number with a different number of ‘2’s you’ll get a different number. Two ‘2’s and a ‘3’ gives you 2 x 2 x 3 = 12, whereas three ‘2’s and a ‘3’ gives you 2 x 2 x 2 x 3 = 24. Zero really doesn’t work as a building block. It acts more like a stick of dynamite that blows away all your other building blocks whenever you try to multiply by it.
We’re almost ready to restate the fancy definition of prime numbers at the top of the page. But, first, there’s one more thing that keeps coming up – words like “whole numbers.”
There’s different kinds of numbers?
We keep insisting that prime and composite numbers be whole numbers. There’s different kinds of numbers that are useful for different things. If that sounds odd (strange) then read about the different kinds of numbers.
Imagine if rational numbers (the ones that include fractions) could be prime. Talking about them would be very boring. The reason is that every rational number, except zero, is a factor of every other rational number. That would mean that all numbers would be composite, and there would not be any such thing as a prime number! To avoid this, some definitions of prime numbers say that they are counting numbers (the ones you use to count sheep), and others say that they are integers (the ones that can be positive and negative).
In fact, it doesn’t make a difference whether the definitions say that they are counting numbers, natural numbers, whole numbers, or integers. The reason is that they also should say “must be greater than one,” which rules out negative numbers, zero, and one. Negative numbers and zero are the difference between counting numbers, natural numbers, and the integers. That’s why we’ve been intentionally saying “whole numbers” until now, because everyone knows more or less what that means, and we didn’t want to get lost in a big long discussion of the different kinds of numbers until now.
OK! Now that we’ve gone over the important side-topics in detail, let’s ask the original question again.
What is a Prime Number?
Well, we’d like to say “Prime numbers are numbers that aren’t composite.” But, Oopsie Doodles! That would ignore those annoying little details and our wise-guy friends would give us a hard time whining “one and a half isn’t composite, so isn’t it a prime number?”. Let’s fix it, with a good, simple definition:
- Prime numbers are whole numbers greater than one that are not composite.
- Composite numbers are products of whole numbers greater than one.
Why are prime numbers important?
There’s lots of answers to this question depending on whether you’d like to know practical reasons to care about prime numbers, or deeper reasons.
Prime numbers, and practical math:
Some of the practical things prime numbers are good for include:
- Simplifying fractions
- finding LCDs (Least Common Denominators)
- finding GCFs (Greatest Common Factors)
Just finding giant prime numbers is important. It’s a huge amount of work to find out if really big numbers are prime. For that reason, prime numbers are important in cryptology, which is the study of how to make secret codes. They are used for such practical things as keeping bank accounts secure. Prime numbers are big money!!!
The Riemann Hypothesis
Prime numbers also have to do with some of the most interesting problems in mathematics that haven’t been solved yet. Math people call unsolved problems “open problems.” The most famous one today is probably the Riemann Hypothesis. It takes a lot of explaining to understand, and really, no-one has truly got to the bottom of it yet. It has to do with guessing how likely a number is to be prime based on how big the number is. We can’t explain much more about it here, although we can recommend a book about it for people who aren’t mathematicians. By the way, there’s big prizes waiting for anyone who can solve important open problems, including the Riemann Hypothesis.
Why are prime numbers defined the way they are?
Good question! Often people just accept definitions the way they’re given without wondering why they were defined the way they are. After all definitions can be made however we want as long as we accept the consequences of our choices. So — why exclude the number one from being a prime number? Why not negative numbers? The reason is that by defining the prime numbers the way we do — without negative numbers, zero, or one — we can state an amazing property of whole numbers in a simple way: The Fundamental Theorem of Arithmetic.
Next Topic: The Fundamental Theorem of Arithmetic