Title: TTK-005-Equal Complex Numbers

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Equal Complex Numbers

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Note.

If two real numbers are not equal, then these numbers can always be compared and sorted by size. However, complex invoices cannot be sorted by size.

For example, you cannot determine which complex number is larger, either \(2 + 3\cdot i\) or \(3 + 2\cdot i\).

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Two complex numbers are equal if their real parts are the same and their imaginary parts are the same.

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\(\color{orange}{a}+\color{green}{b}i=\color{orange}{c}+\color{green}{d}i\;\;\Leftrightarrow\;\; \color{orange}{a}=\color{orange}{c} \;\;\text{and}\;\; \color{green}{b}=\color{green}{d}\)

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Example 1

not equal

equal

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Example 2 Find the value of \(x\) so that the complex numbers \(3\! -\! 5i\) and \(3\!-\!xi\) are equal.

Solution: Since \(3 - 5i=3-xi\), then \(-5=-x\) and \(x=5\)

Conjugate Fill in the Blanks

Find the values of \(x\) and \(y\) so that the complex numbers \(x+yi\) and \(2y+(x-2)i\) are equal.

Q: Answer: \(x=\)[...]  and \(y=\)[...]
A: 4; 2
Feedback (0 to 100%): Solution: \(\Rightarrow \begin{cases}x = 2y\\y = x-2\end{cases}\Rightarrow \begin{cases}x = 2y\\y = 2y-2\end{cases} \Rightarrow \begin{cases}x = 4\\y = 2\end{cases}\)

Võrdsed kompleksarvud 1

Find \(x\) and \(y\) values ​​so that the complex numbers \(0.5x-2i\) and \( -6+yi\) are equal.

Q: Answer: \(x=\)[...], \(y=\)[...] .
A: -12; -2
Feedback (0 to 100%): Solution: \( 0.5x-2i = -6+yi\Rightarrow \begin{cases}0.5x = -6\\-2 = y\end{cases}\Rightarrow \begin{cases}x = -12\\y = -2\end{cases}\)

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Exercise