5.8: Defining Complex Numbers
The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?
Watch This
First, watch this video.
Khan Academy: Introduction to i and Imaginary Numbers
Then, watch this video.
Guidance
Before this concept, all numbers have been real numbers. 2, -5,
In order to take the square root of a negative number we are going to assign
All complex numbers have the form
Example A
Find
Solution: First pull out the
Investigation: Powers of i
In addition to now being able to take the square root of a negative number,
1. Write out
2. Write out
3. Write out
4. Write out
5. Write out
6. Do you see a pattern? Describe it and try to find
You should see that the powers of
Example B
Find:
a)
b)
c)
Solution:
a) 32 is divisible by 4, so
b)
c)
Example C
Simplify the complex expressions.
a) \begin{align*}(6-4i)+(5+8i)\end{align*}
b) \begin{align*}9-(4+i)+(2-7i)\end{align*}
Solution: To add or subtract complex numbers, you need to combine like terms. Be careful with negatives and properly distributing them. Your answer should always be in standard form, which is \begin{align*}a + bi\end{align*}.
a) \begin{align*}(6-4i)+(5+8i)={\color{red}6}{\color{blue}-4i}+{\color{red}5}+{\color{blue}8i}={\color{red}11}+{\color{blue}4i}\end{align*}
b) \begin{align*}9-(4+i)+(2-7i)={\color{red}9-4}{\color{blue}-i}+{\color{red}2}{\color{blue}-7i}={\color{red}7}{\color{blue}-8i}\end{align*}
Intro Problem Revisit We're looking for \begin{align*}\sqrt{-460}\end{align*} .
First we need to pull out the \begin{align*}i\end{align*}. Then, we need to simplify \begin{align*}\sqrt{460}\end{align*} .
\begin{align*}\sqrt{-460}=\sqrt{-1} \cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4 \cdot 115}=2i\sqrt{115}\end{align*}
Guided Practice
Simplify.
1. \begin{align*}\sqrt{-49}\end{align*}
2. \begin{align*}\sqrt{-125}\end{align*}
3. \begin{align*}i^{210}\end{align*}
4. \begin{align*}(8-3i)-(12-i)\end{align*}
Answers
1. Rewrite \begin{align*}\sqrt{-49}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.
\begin{align*}\sqrt{-49}=i\sqrt{49}=7i\end{align*}
2. Rewrite \begin{align*}\sqrt{-125}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.
\begin{align*}\sqrt{-125}=i\sqrt{125}=i\sqrt{25 \cdot 5}=5i\sqrt{5}\end{align*}
3. \begin{align*}210 \div 4=52\end{align*}, with a remainder of 2. Therefore, \begin{align*}i^{210}=i^2=-1\end{align*}.
4. Distribute the negative and combine like terms.
\begin{align*}(8-3i)-(12-i)=8-3i-12+i=-4-2i\end{align*}
Vocabulary
- Imaginary Numbers
- Any number with an \begin{align*}i\end{align*} associated with it. Imaginary numbers have the form \begin{align*}a + bi\end{align*} or \begin{align*}bi\end{align*}.
- Complex Numbers
- All real and imaginary numbers. Complex numbers have the standard form \begin{align*}a + bi\end{align*}, where \begin{align*}a\end{align*} or \begin{align*}b\end{align*} can be zero. \begin{align*}a\end{align*} is the real part and \begin{align*}bi\end{align*} is the imaginary part.
- Pure Imaginary Numbers
- An imaginary number without a real part, only \begin{align*}bi\end{align*}.
Practice
Simplify each expression and write in standard form.
- \begin{align*}\sqrt{-9}\end{align*}
- \begin{align*}\sqrt{-242}\end{align*}
- \begin{align*}6\sqrt{-45}\end{align*}
- \begin{align*}-12i\sqrt{98}\end{align*}
- \begin{align*}\sqrt{-32} \cdot \sqrt{-27}\end{align*}
- \begin{align*}7i\sqrt{-126}\end{align*}
- \begin{align*}i^8\end{align*}
- \begin{align*}16i^{22}\end{align*}
- \begin{align*}-9i^{65}\end{align*}
- \begin{align*}i^{365}\end{align*}
- \begin{align*}2i^{91}\end{align*}
- \begin{align*}\sqrt{-\frac{16}{80}}\end{align*}
- \begin{align*}(11-5i)+(6-7i)\end{align*}
- \begin{align*}(14+2i)-(20+9i)\end{align*}
- \begin{align*}(8-i)-(3+4i)+15i\end{align*}
- \begin{align*}-10i-(1-4i)\end{align*}
- \begin{align*}(0.2+1.5i)-(-0.6+i)\end{align*}
- \begin{align*}6+(18-i)-(2+12i)\end{align*}
- \begin{align*}-i+(19+22i)-(8-14i)\end{align*}
- \begin{align*}18-(4+6i)+(17-9i)+24i\end{align*}
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Color | Highlighted Text | Notes | |
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Show More |
Term | Definition |
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is an imaginary number. . | |
Absolute Value | The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin. |
Complex Conjugate | Complex conjugates are pairs of complex binomials. The complex conjugate of is . When complex conjugates are multiplied, the result is a single real number. |
i | is an imaginary number. . |
Imaginary Number | An imaginary number is a number that can be written as the product of a real number and . |
imaginary part | The imaginary part of a complex number is . |
Pure Imaginary Numbers | The pure imaginary numbers are the subset of complex numbers without real parts, only . |
Real Number | A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers. |
real part | The real part of a complex number is . |
rectangular coordinates | A point is written using rectangular coordinates if it is written in terms of and and can be graphed on the Cartesian plane. |
rectangular form | The rectangular form of a point or a curve is given in terms of and and is graphed on the Cartesian plane. |
standard form | The standard form of a complex number is where and are real numbers. |
Image Attributions
Here you'll define, discover the “powers of @$\begin{align*}i\end{align*}@$,” and add and subtract complex and imaginary numbers.