(Multiplication) If the product of two complex numbers is zero, show that at least one

Show that \(i^{2}=-1, i^{3}=-i, i^{4}=1\), \(i^{5}=i . \cdots\) and \(1 / i=-i .1 / i^{2}=-1,1 / i^{3}=i . \cdots\). Text Transcription: i^2 = -1, i^3 = - i, i^4 = 1, i^{5} = i, \cdots 1/i = - i, 1/i^2 = - 1, 1/i^3 = i, cdots

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\bar{z}_{1} / \bar{z}_{2}, \overline{\left(z_{1} / z_{2}\right)}\) Text Transcription: barz_1/barz_2, overline{(z_1/z_2)}

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\bar{z}_{1} / z_{1}, z_{1} / \bar{z}_{1}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i bar z_1/z_1, z_1/barz_1

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\left(z_{1}+z_{2}\right) /\left(z_{1}-z_{2}\right)\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i (z_1 + z_2)/(z_1 - z_2)

Let z = x + iy. Find: \(\operatorname{Im} z^{3},(\operatorname{Im} z)^{3}\) Text Transcription: Im z^3, (Im z)^3

Let z = x + iy. Find: \(\operatorname{Re}(1 / \bar{z})\) Text Transcription: Re(1/bar z)

Let z = x + iy. Find: \(\operatorname{Im}\left[(1+i)^{8} z^{2}\right]\) Text Transcription: Im[(1 + i)^8 z^2]

Let z = x + iy. Find: \(\operatorname{Re}\left(1 / \bar{z}^{2}\right)\) Text Transcription: Re (1/bar z^2)

Derive the following laws for complex numbers from the corresponding laws for real numbers. \(z_{1}+z_{2}=z_{2}+z_{1}, z_{1} z_{2}=z_{2} z_{1}\) (Commutative laws) \(\left(z_{1}+z_{2}\right)+z_{3}=z_{1}+\left(z_{2}+z_{3}\right)\) (Associative laws) \(\left(z_{1} z_{2}\right) z_{3}=z_{1}\left(z_{2} z_{3}\right)\) (Associative laws) \(z_{1}\left(z_{2}+z_{3}\right)=z_{1} z_{2}+z_{1} z_{3}\) (Distributive law) 0 + z = z + 0 = z, \(z+(-z)=(-z)+z=0, \quad z \cdot 1=z\) Text Transcription: z_1 + z_2 = z_2 + z_1, z_1 z_2 = z_2 z_1 (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) (z_1 z_2) z_3 = z_1 (z_2 z_3) z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3 z+(-z) = (-z) + z = 0, z cdot 1 = z

Multiplication by i is geometrically a counterclockwise rotation through \(\pi / 2\left(90^{\circ}\right)\). Verify this by graphing z and iz and the angle of rotation for z = 2 + 2i, z = - 1 - 5i, z = 4 - 3i. Text Transcription: pi/2 (90^{circ})

Let':l = 2 + 3i and Z2 = 4 - 5i. Showing the details of your work. find (in the form x + iy):

Let':l = 2 + 3i and Z2 = 4 - 5i. Showing the details of your work. find (in the form x + iy):

Show that z = x + iy is pure imaginary if and only if \(\bar{z}\) = - z. Text Transcription: bar z

Verify (9) for \(z_{1}=24+10 i\), \(z_{2}=4+6 i\). Text Transcription: z_1 = 24 + 10i z_2 = 4 + 6i

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\left(5 z_{1}+3 z_{2}\right)^{2}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i (5z_1 + 3z_2)^2

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\bar{z_{1}, \bar{z_{2}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i bar{z_1}, bar{z_2}

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): Re \(\left(1 / z_{1}^{2}\right)\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i (1/z_1^2)

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): Re \(\left(z_{2}^{2}\right)\), (Re \(\bar{z}_{2})^{2}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i (z_2^2) (Re bar z_2)^2

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(z_{2} / z_{1}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i z_2/z_1

Let \(z_{1}=2+3 i\) and \(z_{2}=4-5 i\). Showing the details of your work, find (in the form x + iy): \(\left(4 z_{1}-z_{2}\right)^{2}\) Text Transcription: z_1 = 2 + 3i z_2 = 4 - 5i (4z_1 - z_2)^2