Title: TTK004 Imaginary Numbers

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Imaginary Numbers

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Imaginary numbers is any number that can be written in the form \(\;a\cdot{i}\), where \(a\) is a real number and \(i\) is an imaginary unit (\(i^2=-1\)).

Important!

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Remember that the multiplication sign may not be written in front of the signed factor. So \(\;5i\;\) means the same as \(\;5\cdot i\). Here on the slides, we omit the multiplication sign.

The imaginary unit can also be used to represent the square root of negative numbers, for example

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\(\sqrt{-25}=\sqrt{25\cdot(-1)}=\sqrt{25}\cdot\sqrt{-1}=5\cdot i=5i\)

\(\sqrt{-100}=\sqrt{100\cdot(-1)}=\sqrt{100}\cdot\sqrt{-1}=10i\)

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Example. Imaginary numbers are \(2i\), \(\;-7i\), \(\;\sqrt{2}i\), \(\;\frac{3}{2}i\)

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\(2i\) is imaginary number, where \(2\) is real number and \(i\) is imaginary unit;

\(-7i\) is imaginary number, where \(-7\) is real number and \(i\) is imaginary unit.

Imaginary Numbers Multiple Choice

Q: Choose imaginary numbers
- \(3i\) (correct)
- \(6\)
- \(\sqrt[3]{72}i\) (correct)
- \(-1\)
Feedback (0 to 100%): Solution: An imaginary number is any number that can be written in the form \(\;a\cdot{i}\) where \(a\) is a real number and \(i\) is an imaginary unit.

Imaginary numbers Fill in the Blanks

Calculate

Q: \(\sqrt{-16}\) = [...]

\(\sqrt{-144}\) = [...]
A: 4i; 12i
Feedback (0 to 100%): Solution: \(\sqrt{-16}=\sqrt{16\cdot(-1)}=\sqrt{16}\cdot\sqrt{-1}=4\cdot i=4i\) \(\sqrt{-144}=\sqrt{144\cdot(-1)}=\sqrt{144}\cdot\sqrt{-1}=12i\)

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Exercises