### About Absolute Value of a Complex Number:

The absolute value of a complex number is the distance from the origin (0 + 0i) to the complex number (a + bi) on the complex plane. To find this measure, we can think about the distance formula that was used previously in the course. We know that the distance between any two points, labeled as d, is given as follows: d = sqrt[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]. If we replace d with the absolute value of a + bi, we obtain: |a + bi| = sqrt[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}] Next we can think about the fact that we are now dealing with real values in the place of x-values and imaginary values in the place of y-values. Since one real value will be a and the other will be zero, we can rewrite the formula as: |a + bi| = sqrt[a^{2} + (y_{2} - y_{1})^{2}]. Similarly, we find that one imaginary value will be b and the other will be zero, again, we can rewrite the formula as: |a + bi| = sqrt[a^{2} + b^{2}]. So now we see that the absolute value of any complex number is given as the square root of a, the real part squared, plus b, the imaginary part squared.

Test Objectives

- Demonstrate the ability to find the absolute value of a complex number

#1:

Instructions: find the absolute value of each.

$$a)\hspace{.2em}|-3 + 3i|$$

$$b)\hspace{.2em}|-i|$$

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#2:

Instructions: find the absolute value of each.

$$a)\hspace{.2em}|1 + 2i|$$

$$b)\hspace{.2em}|3i|$$

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#3:

Instructions: find the absolute value of each.

$$a)\hspace{.2em}|-3i|$$

$$b)\hspace{.2em}|3 + 5i|$$

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#4:

Instructions: find the absolute value of each.

$$a)\hspace{.2em}|2 + 6i|$$

$$b)\hspace{.2em}|-8 - 8i|$$

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#5:

Instructions: find the absolute value of each.

$$a)\hspace{.2em}|6 + 3i|$$

$$b)\hspace{.2em}|-4 + 4i|$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}3\sqrt{2}$$

$$b)\hspace{.2em}1$$

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#2:

Solutions:

$$a)\hspace{.2em}\sqrt{5}$$

$$b)\hspace{.2em}3$$

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#3:

Solutions:

$$a)\hspace{.2em}3$$

$$b)\hspace{.2em}\sqrt{34}$$

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#4:

Solutions:

$$a)\hspace{.2em}2\sqrt{10}$$

$$b)\hspace{.2em}8\sqrt{2}$$

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#5:

Solutions:

$$a)\hspace{.2em}3\sqrt{5}$$

$$b)\hspace{.2em}4\sqrt{2}$$