Historia om tala

• \(\mathbb{N} = \{ 1,2,3,\ldots \}\)
  • \(0\)
  • \(-1,-2,-3,\ldots\)
• \(\mathbb{Z} = \{ \ldots, -3,-2,-1,0,1,2,3,\ldots \}\)
• \(\mathbb{Q} = \{ \frac nm | n\in\mathbb{Z}, m\in\mathbb{N} \}\)

$$i^2 = -1$$

$$i = \sqrt{-1}$$

$$x^2 + 2x + 2 = 0$$

$$ax^2 + bx + c = 0 $$

$$ \begin{align} \begin{split} x & = \frac{-b \pm\sqrt{b^2 - 4ac}}{2a} \\& = \frac{-2 \pm\sqrt{2^2 - 4\cdot2}}{2} \\& = \frac{-2 \pm\sqrt{4 - 8}}{2} \\& = \frac{-2 \pm 2\sqrt{-1}}{2} \\& = \frac{-2 \pm 2i}{2} = -1 \pm i \end{split} \end{align} $$

$$ \begin{align} \begin{split} (-1 - i)^2 + 2(-1 - i) + 2 & = (1 + 2i + i^2) + (-2 - 2i) + 2 & = 0 \end{split} \end{align} $$