{3^3 , 5, 7^2 }. That’s okay if you can remember that 3^3 does not literally mean 27 in this context, and 7^2 does
not literally mean 49. (Neither 27 nor 49 are prime!)
The clearest way to express the prime factors of a composite number is to write out the product, list-
ing each factor as many times as it “deserves,” and using multiplication symbols between them. You can
arrange the product in any order, but it helps if you start with the smallest factor and go up, or start with
the largest factor and go down.
There is only one way to factor a composite number into a product of primes. This fact is called the
Fundamental Theorem of Arithmetic.Here’s a challenge!
The numbers 2 and 3 are both prime, and they are also consecutive whole numbers. Are there any other
examples of two consecutive whole numbers that are both prime?Solution
No, none of the even numbers larger than 2 (i.e., 4, 6, 8, 10, etc.) are prime. We can factor 2 out of any
such number, and we always get a natural number bigger than 1. By elimination, then, all the primes larger
than 2 are odd. If we take any of these primes and then find the next natural number, we are adding 1 to
an odd number. That always produces an even number, which we have just seen can’t be prime.
The conclusion: when we come across any prime number larger than 2, the next consecutive natural
number is always composite.The Integers
Centuries ago, negative numbers weren’t taken seriously. How could you have less than none
of anything? When the set of all negative whole numbers was finally joined together with the
set of natural numbers, the result became known as the set of integers. That set is symbolized
Z, like this:Z= {..., −3, −2, −1, 0, 1, 2, 3, ...}Negative numbers
What is a negative number, exactly? That question is deeper than it seems at first thought. We
can start to answer it by creating situations where negative numbers are really useful.
In the United States, most nonscientific people use the Fahrenheit temperature scale, where
32 degrees represents the freezing point of water. Scientists, and people outside the United
States, use the Celsius temperature scale, where 0 degrees represents the freezing point of water.
In either system, temperatures often get colder than 0 degrees. Then people start calling tem-
peratures negative.
Here’s another real-life situation where negative numbers come in handy. These days,
nearly everyone has a credit card. When you first get the card, it has a balance of 0. That
means you haven’t put any money in the bank that gave you the card, but you don’t owe the
bank any money, either. What if you buy some items at the local department store, “charging”
up a balance of $49? How much money is in the account now? If you think of it as the bank’sThe Integers 45