You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:

Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$

$$ \begin{array}{cc}

n & s(n) \\

1 & 1 \\

2 & 2 \\

3 & 3 \\

4 & 4 \\

\cdots & \cdots \end{array} $$

Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$

$$ \begin{array}{cc}

n & s(n) \\

1 & 1 \\

2 & -1 \\

3 & 2 \\

4 & -2 \\

\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.

By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).

Wonderful blog and nice explanation,I am here to discuss something about natural numbers.Natural numbers is the set of non-negative integers and this includes zero as a natural number.

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