Property 1
\(x^2=a,\ a\in\mathbb R^-\)
\(x=\sqrt a,\ x\notin\mathbb R\)
\(i^2=-1\qquad\sqrt{-1}=i\)
For \(a,b\in\mathbb R^+\), \(\sqrt a\cdot\sqrt b=\sqrt{ab}\). For negative real numbers this rule is not used in that form.
Example: \(\sqrt{-4}\cdot\sqrt{-1}=2i^2=-2\)
1.
\(\sqrt{-9}=?\)
2.
\(\sqrt{-4}=?\)
3.
\(\sqrt{-1}=?\)
4.
\(\sqrt{-25}=?\)
5.
\(\sqrt{-32}=?\)
6.
\(\sqrt[3]{-8}=?\)
7.
\(\sqrt{-4}\cdot\sqrt9=?\)
8.
\(\sqrt{-4}\cdot\sqrt{-9}=?\)
9.
\(\sqrt{-25}\cdot\sqrt{-1}=?\)
10.
\(\sqrt{-3}\cdot\sqrt{-12}=?\)
Property 2
\(a\neq0,\ a,b,c\in\mathbb R\)
\(ax^2+bx+c=0\)
\(\Delta=b^2-4ac\)
\(\Delta>0\): two different real roots.
\(\Delta=0\): one real root.
\(\Delta<0\): there is no real root; the roots are complex numbers.
1.
\(x\in\mathbb C\)
\(x^2+36=0\)
\(\Rightarrow S.S.=?\)
2.
\(x\in\mathbb C\)
\(x^2+49=0\)
\(\Rightarrow S.S.=?\)
3.
\(x\in\mathbb C\)
\(x^2-2x+5=0\)
\(\Rightarrow S.S.=?\)
4.
\(x\in\mathbb C\)
\(x^2-6x+10=0\)
\(\Rightarrow S.S.=?\)
5.
\(x\in\mathbb C\)
\(x^2-4x+5=0\)
\(\Rightarrow S.S.=?\)
6.
\(x\in\mathbb C\)
\(x^2-8x+17=0\)
\(\Rightarrow S.S.=?\)
7.
\(x\in\mathbb C\)
\(x^2-4x+8=0\)
\(\Rightarrow S.S.=?\)
Property 3
\(z\in\mathbb C,\ a,b\in\mathbb R\)
\(z=a+ib\)
\(\operatorname{Re}(z)=a\)
\(\operatorname{Im}(z)=b\)
1.
\(z=2-i\)
\(\Rightarrow\operatorname{Im}(z)=?\)
2.
\(z=3+4i\)
\(\Rightarrow\operatorname{Re}(z)=?\)
3.
\(z=3i-2\)
\(\Rightarrow\operatorname{Re}(z)=?\)
4.
\(z=6i\)
\(\Rightarrow\operatorname{Re}(z)=?\)
5.
\(z=\sqrt{-25}+9\)
\(\Rightarrow\operatorname{Im}(z)=?\)
6.
\(z=\sqrt{36}-\sqrt{-25}\)
\(\Rightarrow\operatorname{Im}(z)=?\)
7.
\(z=3-2i\)
\(\Rightarrow\operatorname{Im}(z)\operatorname{Re}(z)=?\)
8.
\(z=1-3i\)
\(\Rightarrow\samefrac{\operatorname{Im}(z)}{\operatorname{Re}(z)}=?\)
Property 4
\(z=a+ib\) is represented by the point \((a,b)\) on the complex plane.
\(a+ib=c+id\iff a=c\) and \(b=d\).
1.
\(x,y\in\mathbb R\)
\(x+yi=2+5i\)
\(\Rightarrow x+y=?\)
2.
\(x,y\in\mathbb R\)
\(x+4+2i=6-(y+1)i\)
\(\Rightarrow xy=?\)
3.
\(a,b\in\mathbb R\)
\(3a-6bi-2=10-18i\)
\(\Rightarrow ab=?\)
4.
\(x,y\in\mathbb R\)
\(2x+3yi=2y+4-6i\)
\(\Rightarrow x=?\)
5.
\(x,y\in\mathbb R\)
\(2x+1+4yi=5+8i\)
\(\Rightarrow x+y=?\)
6.
\(x,y\in\mathbb R\)
\(x^2-4+(y+3)i=(6+x)i\)
\(\Rightarrow\sum y=?\)
7.
\(x,y\in\mathbb R\)
\(3x-y+5i=7+(x+y)i\)
\(\Rightarrow x=?\)
8.
\(x,y\in\mathbb R\)
\(3x+2yi=2y+(2x+4)i\)
\(\Rightarrow x-y=?\)