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=NThis 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 = 99Special Natural Numbers 39