Hello. I'm Professor Von Schmohawk and welcome to Why U.
Of the four basic arithmetic operations
so far we have considered addition
subtraction, which is like the addition of negative numbers
and multiplication which is like repeated addition.
The fourth arithmetic operation is division.
Division is the inverse of multiplication
since it undoes the result of a multiplication operation.
For example, multiplying two by three gives us six.
And dividing six by three gives us back our original two
undoing the multiplication.
There are two ways to interpret what a division operation represents.
Take, for example, six divided by three.
One way to interpret this is that we are to divide six into three equal pieces.
Each piece will end up having a length of two.
A second and equally legitimate way to interpret six divided by three
is that we are asking how many times three fits into six.
The answer is that three fits into six two times.
Of course, no matter which interpretation we prefer, six divided by three is two.
In this example, we divided the natural number six by the natural number three
and we got another natural number, two.
But the result of a division operation involving natural numbers
cannot always be represented by a natural number.
For example, what if we divide five by two?
Using the first interpretation we divide five into two equal pieces.
The length of each piece is greater than two but less than three
so this length cannot be represented by a natural number.
Using the alternate interpretation, we ask how many times two fits into five.
The answer is that two does not fit into five evenly.
Either way, the result of five divided by two
cannot be represented by a natural number.
Let’s see if there are any natural numbers which fit into five evenly.
One fits into five exactly five times.
We already saw that two doesn’t fit evenly.
Nor does three
or four.
But five fits into five exactly one time.
In fact, any natural number can be divided by one or itself evenly.
Numbers that can divide evenly into a natural number
are said to be that number's "divisors" or "factors".
Since three can divide evenly into six, we say that three is a factor of six.
In fact, six has four natural number factors, one, two, three, and six.
On the other hand, five has only two factors, one and five.
And one has only one factor, itself.
Natural numbers that have exactly two factors, themselves and one, are called "prime numbers".
So which natural numbers are prime?
Let's look at the first ten natural numbers.
As we saw, five has exactly two natural divisors so five is a prime number.
The same thing goes for two, three, and seven.
However, the number one has only a single divisor, itself, so one is not a prime number.
Natural numbers which have more than two divisors are called "composite numbers".
All composite numbers can be broken down into a product of prime numbers.
For instance, the composite number four
is the product of the prime number two times itself.
Six is a product of the primes two and three.
Eight is two times two times two.
Nine is three times three.
And ten is two times five.
To determine which natural numbers are prime
we can use a process invented by the ancient Greek mathematician Eratosthenes.
This process, called the "Sieve of Eratosthenes"
involves eliminating all composite numbers from a list of natural numbers.
The numbers that are left are prime.
Let’s try it for the natural numbers two through one-hundred.
We start by marking two as the first prime number.
Then we eliminate all numbers divisible by two from our list
since these are composite numbers.
Once we eliminate these numbers, the next number in our list, three, will be prime.
We then eliminate all the numbers divisible by three, since these are composite numbers.
Now, the next number in our list, five, will be prime.
We then eliminate all the numbers divisible by five.
Now, the next number in our list, seven, will be prime.
We then eliminate all the numbers divisible by seven.
We can stop testing prime numbers once we reach a number which, when multiplied by itself
is greater than the highest number remaining in our list.
The next number, eleven, multiplied by itself is greater than 97
so we can stop testing.
We have eliminated all the composite numbers.
Any remaining numbers are prime.
It is sometimes hard to tell whether a large number is prime.
However, there are a few easy tricks for spotting some numbers which are composite
and therefore not prime.
First of all, all even numbers are composite except for two
since even numbers are all divisible by two.
Therefore any number greater than two that ends with 0, 2, 4, 6, or 8
is a composite number.
For example, all of these numbers are composite
since they end with 0, 2, 4, 6, or 8 and are therefore divisible by two.
Also, any number which ends with 5 or 0 is divisible by five, and is thus composite.
In addition, there is an interesting trick
for determining if a number is evenly divisible by three.
A number is divisible by three if the sum of its digits is divisible by three.
For example, the number 1761
is composed of digits 1, 7, 6, and 1 whose sum is fifteen.
Since fifteen is evenly divisible by three then so is 1761.
There are many tricks for testing whether a number is divisible by other primes.
For instance, to determine if a number is divisible by eleven
first separate the last digit from the rest of the digits in the number.
Then subtract the last digit from the number formed by the other digits.
If the result is divisible by eleven then so is the original number.
If it is not obvious if the result is divisible by eleven
then the process can be repeated with the number formed by the other digits.
There are similar techniques for determining if a number is divisible by other primes
like seven, thirteen, seventeen, nineteen and so on.
Since it can be extremely difficult to tell whether a very large number is prime
some mathematicians compete to break the record
for finding a prime number larger than anyone else has found.
There is no "largest" prime number
so this contest will likely go on as long as there are mathematicians
with a lot of time on their hands.