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A “complex number” is the sum of two parts: a real number by itself, and a real number multiplied by $i$ . It can therefore be written as $a+\mathrm{bi}$ , where $a$ and $b$ are real numbers.
The first part, $a$ , is referred to as the real part . The second part, $\mathrm{bi}$ , is referred to as the imaginary part .
Examples of complex numbers $a+\mathrm{bi}$ ( $a$ is the “real part”; $\mathrm{bi}$ is the “imaginary part”) | |
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$3+\mathrm{2i}$ | $a=3$ , $b=2$ |
$\mathrm{\pi}$ | $a=\mathrm{\pi}$ , $b=0$ (no imaginary part: a “pure real number”) |
$\mathrm{-i}$ | $a=0$ , $b=-1$ (no real part: a “pure imaginary number”) |
Some numbers are not obviously in the form $a+\mathrm{bi}$ . However, any number can be put in this form.
Example 1: Putting a fraction into $a+\mathrm{bi}$ form ( $i$ in the numerator) |
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$\frac{3-\mathrm{4i}}{5}$ is a valid complex number. But it is not in the form $a+\mathrm{bi}$ , and we cannot immediately see what the real and imaginary parts are. |
To see the parts, we rewrite it like this: |
$\frac{3-\mathrm{4i}}{5}$ $=$ $\frac{3}{5}$ – $\frac{4}{5}i$ |
Why does that work? It’s just the ordinary rules of fractions, applied backward. (Try multiplying and then subtracting on the right to confirm this.) But now we have a form we can use: |
$\frac{3-\mathrm{4i}}{5}$ $a=$ $\frac{3}{5}$ , $b=\u2013\frac{4}{5}$ |
So we see that fractions are very easy to break up, if the $i$ is in the numerator. An $i$ in the denominator is a bit trickier to deal with. |
Example 2: Putting a fraction into $a+\mathrm{bi}$ form ( $i$ in the denominator) | |
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$\frac{1}{i}$ $=$ $\frac{1\cdot i}{i\cdot i}$ | Multiplying the top and bottom of a fraction by the same number never changes the value of the fraction: it just rewrites it in a different form. |
$=$ $\frac{i}{-1}$ | Because $i\u2022i$ is ${i}^{2}$ , or –1. |
$=\mathrm{-i}$ | This is not a property of $i$ , but of –1. Similarly, $\frac{5}{-1}$ $=\mathrm{\u20135}$ . |
$\frac{1}{i}$ : $a=0$ , $b=-1$ | since we rewrote it as $\mathrm{-i}$ , or $0-\mathrm{1i}$ |
Finally, what if the denominator is a more complicated complex number? The trick in this case is similar to the trick we used for rationalizing the denominator: we multiply by a quantity known as the complex conjugate of the denominator .
Here is how we can use the “complex conjugate” to simplify a fraction.
Example: Using the Complex Conjugate to put a fraction into $a+\mathrm{bi}$ form | |
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$\frac{5}{3-\mathrm{4i}}$ | The fraction: a complex number not currently in the form $a+bi$ |
$=$ $\frac{5(3+\mathrm{4i})}{(3-\mathrm{4i})(3+\mathrm{4i})}$ | Multiply the top and bottom by the complex conjugate of the denominator |
$=$ $\frac{\text{15}+\text{20}i}{{3}^{2}-(\mathrm{4i}{)}^{2}}$ | Remember, $(x+a)(x\u2013a)={x}^{2}{\mathrm{\u2013a}}^{2}$ |
$=$ $\frac{\text{15}+\text{20}i}{9+\text{16}}$ | ${\left(\mathrm{4i}\right)}^{2}={4}^{2}{i}^{2}=16\left(\mathrm{\u20131}\right)=\mathrm{\u201316}$ , which we are subtracting from 9 |
$=$ $\frac{\text{15}+\text{20}i}{\text{25}}$ | Success! The top has $i$ , but the bottom doesn’t. This is easy to deal with. |
$=$ $\frac{\text{15}}{\text{25}}$ $+$ $\frac{\text{20}i}{\text{25}}$ | Break the fraction up, just as we did in a previous example. |
$=$ $\frac{3}{5}$ $+$ $\frac{4}{5}i$ | So we’re there! $a=$ $\frac{3}{5}$ and $b=\frac{4}{5}$ |
Any number of any kind can be written as $a+\mathrm{bi}$ . The above examples show how to rewrite fractions in this form. In the text, you go through a worksheet designed to rewrite $\sqrt[3]{-1}$ as three different complex numbers. Once you understand this exercise, you can rewrite other radicals, such as $\sqrt{i}$ , in $a+\mathrm{bi}$ form.
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