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Study Guides > Intermediate Algebra

Imaginary and Complex Numbers

Learning Outcomes

  • Express roots of negative numbers in terms of i
  • Express imaginary numbers as bi and complex numbers as a+bia+bi
You really need only one new number to start working with the square roots of negative numbers. That number is the square root of 1,1−1,\sqrt{-1}. The real numbers are those that can be shown on a number line—they seem pretty real to us! When something is not real, we often say it is imaginary. So let us call this new number i and use it to represent the square root of 1−1.

i=1 i=\sqrt{-1}

Because xx=x \sqrt{x}\,\cdot \,\sqrt{x}=x, we can also see that 11=1 \sqrt{-1}\,\cdot \,\sqrt{-1}=-1 or ii=1 i\,\cdot \,i=-1. We also know that ii=i2 i\,\cdot \,i={{i}^{2}}, so we can conclude that i2=1 {{i}^{2}}=-1.

i2=1 {{i}^{2}}=-1

The number i allows us to work with roots of all negative numbers, not just 1 \sqrt{-1}. There are two important rules to remember: 1=i \sqrt{-1}=i, and ab=ab \sqrt{ab}=\sqrt{a}\sqrt{b}. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times 1 \sqrt{-1}. Next you will simplify the square root and rewrite 1 \sqrt{-1} as i. Let us try an example.

Example

Simplify. 4 \sqrt{-4}

Answer: Use the rule ab=ab \sqrt{ab}=\sqrt{a}\sqrt{b} to rewrite this as a product using 1 \sqrt{-1}. 4=41=41 \sqrt{-4}=\sqrt{4\cdot -1}=\sqrt{4}\sqrt{-1} Since 44 is a perfect square (4=22)(4=2^{2}), you can simplify the square root of 44. 41=21 \sqrt{4}\sqrt{-1}=2\sqrt{-1} Use the definition of i to rewrite 1 \sqrt{-1} as i. 21=2i 2\sqrt{-1}=2i The answer is 2i2i.

Example

Simplify. 18 \sqrt{-18}

Answer: Use the rule ab=ab \sqrt{ab}=\sqrt{a}\sqrt{b} to rewrite this as a product using 1 \sqrt{-1}. 18=181=181 \sqrt{-18}=\sqrt{18\cdot -1}=\sqrt{18}\sqrt{-1} Since 1818 is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. In this case, 99 is the only perfect square factor, and the square root of 99 is 33. 181=921=321 \sqrt{18}\sqrt{-1}=\sqrt{9}\sqrt{2}\sqrt{-1}=3\sqrt{2}\sqrt{-1} Use the definition of i to rewrite 1 \sqrt{-1} as i. 321=32i=3i2 3\sqrt{2}\sqrt{-1}=3\sqrt{2}i=3i\sqrt{2} Remember to write i in front of the radical. The answer is 3i23i\sqrt[{}]{2}.

Example

Simplify. 72 -\sqrt{-72}

Answer: Use the rule ab=ab \sqrt{ab}=\sqrt{a}\sqrt{b} to rewrite this as a product using 1 \sqrt{-1}. 72=721=721 -\sqrt{-72}=-\sqrt{72\cdot -1}=-\sqrt{72}\sqrt{-1} Since 7272 is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. Notice that 7272 has three perfect squares as factors: 4,94, 9, and 3636. It is easiest to use the largest factor that is a perfect square. 721=3621=621 -\sqrt{72}\sqrt{-1}=-\sqrt{36}\sqrt{2}\sqrt{-1}=-6\sqrt{2}\sqrt{-1} Use the definition of i to rewrite 1 \sqrt{-1} as i. 621=62i=6i2 -6\sqrt{2}\sqrt{-1}=-6\sqrt{2}i=-6i\sqrt{2} Remember to write i in front of the radical. The answer is 6i2-6i\sqrt[{}]{2}

You may have wanted to simplify 72 -\sqrt{-72} using different factors. Some may have thought of rewriting this radical as 98 -\sqrt{-9}\sqrt{8}, or 418 -\sqrt{-4}\sqrt{18}, or 612 -\sqrt{-6}\sqrt{12} for instance. Each of these radicals would have eventually yielded the same answer of 6i2 -6i\sqrt{2}. In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand. https://youtu.be/LSp7yNP6Xxc

Rewriting the Square Root of a Negative Number

  • Find perfect squares within the radical.
  • Rewrite the radical using the rule ab=ab \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}.
  • Rewrite 1 \sqrt{-1} as i.
Example: 18=92=921=3i2 \sqrt{-18}=\sqrt{9}\sqrt{-2}=\sqrt{9}\sqrt{2}\sqrt{-1}=3i\sqrt{2}

Complex Numbers

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written + bi where a is the real part and bi is the imaginary part. For example, 5+2i5+2i is a complex number. So, too, is 3+4i33+4i\sqrt{3}.

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers. You can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You will see more of that later.

Complex Number Real Part Imaginary Part
3+7i3+7i 33 7i7i
1832i18–32i 1818 32i−32i
35+i2 -\frac{3}{5}+i\sqrt{2} 35 -\frac{3}{5} i2 i\sqrt{2}
2212i \frac{\sqrt{2}}{2}-\frac{1}{2}i 22 \frac{\sqrt{2}}{2} 12i-\frac{1}{2}i
In a number with a radical as part of b, such as 35+i2-\frac{3}{5}+i\sqrt{2} above, the imaginary i should be written in front of the radical. Though writing this number as 35+2i -\frac{3}{5}+\sqrt{2}i is technically correct, it makes it much more difficult to tell whether i is inside or outside of the radical. Putting it before the radical, as in 35+i2 -\frac{3}{5}+i\sqrt{2}, clears up any confusion. Look at these last two examples.
Number Complex Form: a+bia+bi Real Part Imaginary Part
1717 17+0i17+0i 1717 0i0i
3i−3i 03i0–3i 00 3i−3i
By making b=0b=0, any real number can be expressed as a complex number. The real number a is written as a+0ia+0i in complex form. Similarly, any imaginary number can be expressed as a complex number. By making a=0a=0, any imaginary number bibi can be written as 0+bi0+bi in complex form.

Example

Write 83.683.6 as a complex number.

Answer: Remember that a complex number has the form a+bia+bi. You need to figure out what a and b need to be. a+bia+bi Since 83.683.6 is a real number, it is the real part (a) of the complex number a+bia+bi 83.6+bi83.6+bi A real number does not contain any imaginary parts, so the value of bb is 00. The answer is 83.6+0i83.6+0i.

Example

Write 3i−3i as a complex number.

Answer: Remember that a complex number has the form a+bia+bi. You need to figure out what a and b need to be. a+bia+bi Since 3i−3i is an imaginary number, it is the imaginary part bibi of the complex number a+bia+bi. a3ia–3i This imaginary number has no real parts, so the value of aa is 00. The answer is 03i0–3i.

In the next video, we show more examples of how to write numbers as complex numbers. https://youtu.be/mfoOYdDkuyY

Summary

Complex numbers have the form a+bia+bi, where a and b are real numbers and i is the square root of 1−1. All real numbers can be written as complex numbers by setting b=0b=0. Imaginary numbers have the form bi and can also be written as complex numbers by setting a=0a=0. Square roots of negative numbers can be simplified using 1=i \sqrt{-1}=i and ab=ab \sqrt{ab}=\sqrt{a}\sqrt{b}.

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