{∅} {{∅}}
{{{∅}}}
{{{{∅}}}}
{{{{{∅}}}}}
↓ and so on, forever
Why can’t we do that to construct the natural numbers? Well, we could, perhaps. But this method doesn’t increase the number of elements in each set as the defined number gets bigger. Each of the above sets, except for the first, contains one element. To make a good definition of the natural numbers, number theorists want to have the sets contain more and more elements as the defined numbers get larger.
Here’s a challenge!
Write down the natural numbers 0 through 4 purely in terms of braces and the null set symbol, showing how each number can be assembled from “multiple nothings.”
Solution
Start with 0, which is equal to ∅. It helps if you write the numbers as sets of smaller numbers and then break those expressions down:
0= ∅ 1= {0} = {∅}
2= {0, 1} = {∅, {∅}}
3= {0, 1, 2} = {∅, {∅}, {∅, {∅}}}
4= {0, 1, 2, 3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}
Special Natural Numbers
You can classify natural numbers in various ways, just as you can classify people according to blood type, postal zone of residence, country of residence, or even (maybe someday) planet of residence. Here are a few of the most well-known types of natural numbers.
38 Natural Numbers and Integers
Even numbers
Aneven natural number is a whole number whose numeral ends in 0, 2, 4, 6, or 8. If you multiply every number in the set N by 2, you get the set Neven of all the even natural numbers.
This is the familiar set
Neven= {0, 2, 4, 6, 8, 10, ...}
= {0×2, 1×2, 2×2, 3×2, 4×2, 5×2, ...}
Now do the same thing, but backwards. If you divide every number in the set Neven by 2, you get the set N of all natural numbers:
N= {0, 1, 2, 3, 4, 5, ...}
= {0/2, 2/2, 4/2, 6/2, 8/2, 10/2, ...}
Odd numbers
Anodd natural number is a whole number whose numeral ends in 1, 3, 5, 7, or 9. If you mul-tiply every number in the set N by 2 and then add 1 to the result, you get the set Nodd of all the odd natural numbers:
Nodd= {1, 3, 5, 7, 9, 11, ...}
= {(0×2)+1, (1×2)+1, (2×2)+1, (3×2)+1, (4×2)+1, (5×2)+1, ...}
Now do the same thing, but backwards. If you subtract 1 from every number in the set Nodd
and then divide the result by 2, you get the set N of all natural numbers:
N= {0, 1, 2, 3, 4, 5, ...}
= {(1−1)/2, (3−1)/2, (5−1)/2, (7−1)/2, (9−1)/2, (11−1)/2 ...}
The union of the set of all the even natural numbers and the set of all the odd natural numbers is the entire set of natural numbers. You might find this mouthful of words easier to under-stand if you write it in symbols:
Neven∪Nodd=N
This means that you can pick any natural number, as large as you want, and it will always be either even or odd. But you’ll never see a natural number that is both even and odd.
Factors
Whenever you multiply two natural numbers together, you get another natural number. This is obvious, even trivial, when one of the numbers is 0 or 1. Any number times 0 is equal to 0, and any number times 1 is equal to itself. When you start working with the other numbers, things get more interesting.
Let’s pick a natural number, preferably a fairly large one. How about 99? How can we break this number down into a product of other natural numbers besides itself and 1? It’s easy to see that 33 and 3 will work:
33× 3 = 99
Special Natural Numbers 39
40 Natural Numbers and Integers
We can’t break the number 3 down into a product of natural numbers other than itself and 1.
How about 33? This can break down into the product of 11 and 3:
33= 11 × 3
We can’t break 11 down into a product of natural numbers other than itself and 1. Now we have 99 as a product of “unbreakables”:
99= 11 × 3 × 3
Whenever you have a natural number expressed as a product of other numbers, those other numbers are called factors. The process of breaking a number down into a product of other numbers is called factorization or factoring.
Prime and composite numbers
An “unbreakable” natural number is called a prime number, or simply a prime. It’s a natural number larger than 1 that can only be factored into a product of itself and 1. Table 3-1 lists the first 24 prime numbers. If this doesn’t go high enough for you, there are plenty of lists of primes on the Internet.
Any nonprime natural number can be factored into a product of two or more primes.
The numbers in such a product are called the prime factors, and the whole product is called a composite number. When you want to find the prime factors of a large natural number, you can get some help from a calculator that has a square root key. The square root of a number is a smaller number, not always whole, that gives you the original num-ber when multiplied by itself.
Here’s how the process goes. First, use a calculator to find the square root of the number you want to factor. Once you have done that, “chop off ” any nonzero digits that might appear after the decimal point, so you get a whole number. Then add 1 to that number. Call this new whole number s. Now divide the original number by all the primes (referring to Table 3-1) less than or equal to s, starting with the largest prime and working your way down. If you ever get a whole-number quotient as you go through this process, then you know that the divisor and
Table 3-1. The first 24 prime numbers. The number 1 is not considered prime. Any natural number, no matter how large, can be factored into a product of primes.
Order Prime Order Prime Order Prime
1st 2 9th 23 17th 59
the quotient are both factors of the original number. Sometimes the quotient will be prime, and sometimes it won’t be. If it isn’t prime, then it can be factored down further. Don’t stop dividing the original by primes until you get all the way down to 2.
Once you’ve found all the primes smaller than s that divide your original number without giving you a remainder, you have found the prime factors of your original number. Some product of these will give you that original number. You might find that one of the primes is a factor twice, three times, or more. For example:
99= 11 × 3 × 3 297= 11 × 3 × 3 × 3 891= 11 × 3 × 3 × 3 × 3 2,673= 11 × 3 × 3 × 3 × 3 × 3 Perfect squares
Whenever you multiply a number by itself, the process is called squaring, and the product is called a square. If you multiply a natural number by itself, you get a perfect square. If you take the square root of a perfect square, you always get a natural number.
Perfect squares, unlike primes, are easy to find if you have a calculator. Table 3-2 lists the first 24 perfect squares, with 0 and 1 included.
Are you confused?
If you see a large natural number, you can sometimes tell right away that it’s not prime. If its numeral ends in an even number, you know that one of its factors is 2, so it can be factored into a pair of natural numbers other than 1 and itself.
Sometimes, numbers seem at first as if they ought to be prime, but it turns out that they are not. A good example is 39. It can be factored into 13 × 3. Another is 51, which can be factored into 17 × 3. Still another is 57, which can be factored into 19 × 3.
Table 3-2. The first 24 perfect squares. The numbers 0 and 1 are included here.
When you take the square root of a perfect square, you always get a natural number.
Note that we start with the “0th” rather than the “1st” in order here. That way, the order agrees with the number squared.
Order Square Order Square Order Square
0th 0 8th 64 16th 256
Special Natural Numbers 41
42 Natural Numbers and Integers
The best way to find out whether or not a large odd number is prime is to try to factor it into primes.
If the only factors you get are itself and 1 (i.e., if you can’t factor it into primes), then your number is prime. There are some other techniques you can use determine when a number is not prime, such as the
“divisibility” tricks you’ll see later in this chapter.
Here’s a challenge!
When an even number is multiplied by 7, the result always even. Show why this is true.
Solution
For the first few even natural numbers, multiplication by 7 always gives you an even number. Here are the examples for all the single-digit even numbers:
0 × 7 = 0
2 × 7 = 14
4 × 7 = 28
6 × 7 = 42
8 × 7 = 56
You can prove that multiplying any even number by 7 always gives you an even number if you realize that the last digit of an even number is always even. Think of an even number p—any even number. This numberp, however large it might be, must look like one of the following:
______0 ______2 ______4 ______6 ______8
where the long underscore represents any string of digits you want to put there. Now think of “long mul-tiplication” by 7. Remember how you arrange the numerals on the paper and then do the calculations.
You always start out by multiplying the last digits of the two numbers together, getting the last digit of the product. The even number on top, which you are multiplying by the number on the bottom, must end in 0, 2, 4, 6, or 8. If the number on the bottom is 7, then the last digit in the product must be 0, 4 (the second digit in 14), 8 (the second digit in 28), 2 (the second digit in 42), or 6 (the second digit in 56) respectively. The product of any even number and 7 is therefore always even.