Complex Conjugates
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- The complex conjugate of the complex number z=a+biz=a+biz=a+bi, with aaa and bbb real, is z‾=a−bi\overline z=a-biz=a−bi. For example, the complex conjugate of 8+6i8+6i8+6i is 8−…
By Ryan Soedjak, Rajiv Movva, Jonatas Miguel
- The complex conjugate In the previous explanation we described the interchangeability of iii and −i-i−i. Now, let's do it! We replace every iii by −i-i−i: 7i7i7i becomes −7i-7i−…
By Emma Wong, Moor Xu, Victor Wang, Anusha Rahman
- When we have an fraction with iii in both the numerator and the denominator, we want to transform the fraction back to standard form: xi+yxi + yxi+y, where xxx and yyy are real …
By Vivian Loh, Alex Federspiel
- (Video) Complex Conjugates by mathman1024 Here's a video by mathman1024 explaining complex conjugates. A conjugate is when you change the middle of two like terms. So the conjug…
By Alex Federspiel, Anusha Rahman
- Any complex number, zzz, has a conjugate, zˉ\bar{z}zˉ (sometimes written as z∗z^{*}z∗). If we have a complex number in rectangular form, z=a+bi,then,zˉ=a−bi\begin{align}z&=a+bi,…
By Jasper Brown, Alex Federspiel, Anusha Rahman
- Complex Conjugates We are familiar with regular conjugates, and complex conjugates are just a step further, involving imaginary numbers! There are three main characteristics wit…
By Anusha Rahman
Problems 29
- What is the complex conjugate of 20−15i20 - 15i20−15i?
- What is the complex conjugate of −2+3i-2+3i−2+3i?
- What is the complex conjugate of 777?
- What is the complex conjugate of 3i3i3i?
- What is the graphical interpretation (in the complex plane) of a complex conjugate?
- Let zzz be a complex number of magnitude 101010. What is ∣2z‾∣?|2\overline{z}|?∣2z∣?
- What's the product of 1+−51+\sqrt{-5}1+−5 with its complex conjugate?
- What's the product of 1+21+\sqrt{2}1+2 with its complex conjugate? Round your answer to the nearest integer.
- What is the value of the sum ∑z1∣1−z∣2,\sum_z \frac{1}{{\left|1 - z\right|}^2} ,z∑∣1−z∣21, where zzz ranges over all 7 solutions (real and nonreal) of the equation z7=−1z^7 = …
- What is the value of the quotient 2+3i1+6i\frac{2+3i}{1+6i}1+6i2+3i?
- Let aaa and bbb be complex numbers. If ∣a∣=5|a| = 5∣a∣=5 and ∣b∣=7|b| = 7∣b∣=7, what is the value of ∣a‾⋅b‾∣?|\overline{a}\cdot \overline{b}|?∣a⋅b∣?
- Let a=5+12ia = 5 + 12ia=5+12i. What is the value of the sum a+2a‾a + 2\overline{a}a+2a?
- Consider the set of complex numbers z=a+biz = a+biz=a+bi with magnitude ∣z∣=5|z|=5∣z∣=5, such that aaa and bbb are integers. How many pairs of complex conjugates are in this set?
- What is the complex conjugate of x−yix-yix−yi?
- Let z=a+biz=a+biz=a+bi be a complex number. Find z+z‾z+\overline{z}z+z.
- Let z=a+biz=a+biz=a+bi be a complex number. Find z‾−z\overline{z}-zz−z.
- What is the complex conjugate of 8+3i8+3i8+3i?
- What is the complex conjugate of a+bi‾\overline{a+bi}a+bi?
- What is the complex conjugate of ∣6−7i∣\mid 6-7i\mid∣6−7i∣?
- What cubic polynomial has roots 4−i4-i4−i and 333?
- Which of the following are true for complex numbers zzz and www?
- The conjugation operation is which of the following?
- If z2=12+5iz^2=12+5iz2=12+5i, what is zz‾z\overline{z}zz?
- What quadratic polynomial has the root 2+i2+i2+i?
- What cubic polynomial has roots iii and 111?
- What quadratic polynomial has the root −1−5i-1-5i−1−5i?
- If z−z‾=0z-\overline{z}=0z−z=0 for some complex number zzz, which of the following is true?
- Which of the following does zz‾z\overline{z}zz equal?
- Simplify the following for an arbitrary complex number z=a+biz=a+biz=a+bi: z+z‾z−z‾\frac{z+\overline{z}}{z-\overline{z}}z−zz+z