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*}