L'aritmetica dei numeri reali si estende in modo molto semplice agli iperreali; per ogni operazione aritmetica, somma, differenza, prodotto, quoziente, potenza, radicale, logaritmo vale la regola che il risultato è la sequenza dei risultati dell'operazione tra elementi corrispondenti.
Nella tabella qui sotto alcuni esempi.
| Operazioni | ||
|---|---|---|
| Limitati, stabili | $ \begin{array}{r = l} 1 = & \left\langle 1, 1, 1 \cdots 1 \cdots \right\rangle \\ 2 = & \left\langle 2, 2, 2 \cdots 2 \cdots \right\rangle \\ \end{array} $ | $ \begin{array}{r = l} 1 = & \left\langle 1, 1, 1 \cdots 1 \cdots \right\rangle \\ 2 = & \left\langle 2, 2, 2 \cdots 2 \cdots \right\rangle \\ \hline 3 = & \left\langle 3, 3, 3 \cdots 3 \cdots \right\rangle \\ \end{array} $ |
|
Infinitamente piccoli Infinitesimi | $ \begin{array}{r l} \epsilon = & \left\langle 1, \frac{1}{2}, \frac{1}{3} \cdots \frac{1}{n} \cdots \right\rangle = \frac{1}{ω} \\ \alpha = & \left\langle 1, \frac{1}{2}, \frac{1}{4} \cdots \frac{1}{2^n} \cdots \right\rangle \end{array} $ | $ 2ε = \left\langle 2, 1, \frac{2}{3} ... \frac{2}{n} ... \right\rangle = \frac{2}{ω}\\ ζ = \left\langle 1, \frac{1}{3}, \frac{1}{5} ... \frac{1}{2n-1} ... \right\rangle \\ ε + ζ = \left\langle 2, \frac{5}{6}, \frac{8}{15} ... \frac{3n-1}{n(2n-1)} ... \right\rangle \\ ω^ε = \left\langle 1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}} ... n^{\frac{1}{n}} ... \right\rangle \\ $ |
| Limitati | $ \begin{array}{r = l} 1-\alpha = & \left\langle 0, \frac{1}{2}, \frac{3}{4}, \frac{7}{8} \cdots 1- \frac{1}{2^n} \cdots \right\rangle \simeq 1\\ 1+\alpha = & \left\langle 2, \frac{3}{2}, \frac{5}{4}, \frac{9}{8} \cdots 1+ \frac{1}{2^n} \cdots \right\rangle \simeq 1 \end{array} $ | $ \begin{array}{r = l} 1 - \epsilon = & \left\langle 0, \frac{1}{2}, \frac{2}{3} ... \frac{n-1}{n} ... \right\rangle \simeq 1 \\ \eta = & \left\langle 2, \frac{9}{4}, \frac{64}{27} \cdots (1+\epsilon)^{\omega} \cdots \right\rangle \simeq e\\ ωε = & \left\langle 1, 1, 1 ... 1 ... \right\rangle = 1 \\ \frac{ε}{ζ} = & \left\langle 1, \frac{3}{2}, \frac{5}{3} ... \frac{2n-1}{n} = 2-\frac{1}{n} ... \right\rangle = 2 - ε \simeq 2\\ 2^ε = & \left\langle 2, \sqrt[2]{2}, \sqrt[3]{2} ... \sqrt[n]{2} ... \right\rangle \simeq 1 \\ \end{array} $ |
|
Infinitamente grandi Illimitati | $ \begin{array}{r = l} ω = & \left\langle 1, 2, 3 \cdots n \cdots \right\rangle \\ 2ω = & \left\langle 2, 4, 6 \cdots 2n \cdots \right\rangle \\ ω^2 = & \left\langle 1, 4, 9 \cdots n^2 \cdots \right\rangle \\ 2^ω = & \left\langle 2, 4, 8 \cdots 2^n \cdots \right\rangle \\ \end{array} $ |
$ \begin{array}{r = l}
ω = & \left\langle 1, 2, 3, 4 \cdots n \cdots \right\rangle \\
1 = & \left\langle 1, 1, 1, 1 \cdots 1 \cdots \right\rangle \\
\hline
ω-1 = & \left\langle 0, 1, 2, 3 \cdots n+1 \cdots \right\rangle \\
\end{array}
$
$ \sqrt[2]{ω} = ω^{0.5} =\left\langle 1, \sqrt[2]{2}, \sqrt[2]{3} ... \sqrt[2]{n} ... \right\rangle \\ \log_{2}{ω} = \left\langle \log_{2}{1}, \log_{2}{2}, \log_{2}{3} ... \log_{2}{4} ... \right\rangle $ |