square of natural numbers?

Question

consider the natural numbers. each number has a square number.
the squares also have squares. so logically the number of the squares exceeds the actual numbers of the natural numbers?

Answer 1

This is actually an important principal of infinite sets, and you have made a very sophisticated observation. However, Georg Cantor proved that both sets, natural numbers and squares, have the same number of elements (or cardinality), and that number is called aleph-null. There are also aleph-null cubic numbers, even numbers, odd numbers, prime numbers, multiples of 5, and so forth. The essential reason why this is true is because you can set up a one-to-one mapping (a bijective function) between natural numbers and each of these sets. In your example, you pair each natural number with its square. It's okay that the square numbers are paired with their square roots, because there are two separate lists, one for domain (independent variable) and one for range (dependent variable).

Answer 2

Square Of Natural Numbers

Answer 3

The formula for summing the squares of alternate numbers is [n(n + 1)]/2 x (n + 2)/3 This can be rearranged to read 1/6 x n(n + 1)(n + 2) For the sum of the even squares in the range 1 to 41 then n = 40 and the sum = 1/6 x 40 x 41 x 42 = 11,480 I think the formula that you provided is an incomplete version of the sum of ALL square numbers from 1 to n which is 1/6 n(n + 1)(2n +1) NOTE : Various formulae are put forward for summing different combinations of square numbers. I always find it easier to remember that they can all be formed from n(n + 1)/2 which you will probably be familiar with.

Answer 4

Some very complicated answers, but basically, even though each number has a square, and that square has a square, these squares and squares of squares ARE still part of the natural numbers. You haven't found a single number that is not a natural number itself.

Answer 5

First we don't know the exact number of natural numbers. They seem to just keep on increasing forever. Squares too are same. Yes logically we can propose the the number of squares is more than that of natural numbers, but actually we don't exactly know how many of each are present.

Answer 6

No. But consider this: Are there as many points on the number line between 0 and 1 as there are between 0 and 2? I claim there are and I'll prove it as follows:
Let X be the set of all points on the interval [0,1]
Let Y be the set of all points on the interval [0,2]
Now let x ε X. It should be obvious that for all x ε X there is one (and only one) y ε Y such that y=2x. BUt for every y ε Y there also exists one (and only one) x ε X such that x=y/2.

Therefore, the number of points in X and Y are the same since they can be put into a '1-1 correspondence'.

When you start dealing with 'infinities' there are a lot of things which seem to run counter to 'common sense' (an oxymoron if ever there was ☺)


Doug

Answer 7

The set of rational numbers is closed over the operation of Multiplication. I.e the product of any two natural number will be again natural number.

Square is a special case of multiplication operation and hence is still closed over the set of all natural number.

Thus it can be infered that the count of natural numbers is equal to count of there squares since if it was'nt true then the operation of multiplication would'nt have being closed and set {N,x,+} won't have formed a field.

Hope that helps
*<|:-)

Answer 8

You make one incorrect assumption:

"A proper subset of a set is smaller than the original set."

Based on that you conclude that
{ 1, 4, 9, 25, 36, ...} is smaller than { 1, 2, 3, 4, 5, 6, ... }

But it is only true for finite sets. Since { 1, 2, 3, ... } is an infinite set, it need not be the case. Because there is a one-to-one relationship between the sets, they have the same size.

Answer 9

The number would definately be greater because it is the multiple of the number in question.
e.g 5
then 5² = 25
then 25² = 625
∴ 625 exceeds 5

Answer 10

There you are walking on a double edged sowred.

Don't limit yourself , the numbers are infinite so their squares are infinite too.

Limit numbers to 1000 then you see squares or cubes are less in number than normal/natural numbers.