Geometric Representations of Complex Numbers - Algebra 2

What is $\bar{(\frac{z_1}{z_2})}$ for $z_2 \neq 0$ ?

$\bar{(\frac{z_1}{z_2})} = \frac{\bar{z_1}}{\bar{z_2}}$ . Conjugate distributes over division.
What is the geometric meaning of multiplying by $i$ ?

Rotation by $90^circ$ counterclockwise. Multiplication by $i$ rotates by a quarter turn left.
What is the geometric meaning of $z_1z_2$ in polar form?

Multiply moduli; add arguments (scale and rotate). Multiplication scales and rotates simultaneously.
What is the conjugate of $-6+11i$ ?

$-6-11i$ . Negate the imaginary part to find conjugate.
What is the point for $z=-2+7i$ on the complex plane?

$(-2,7)$ . Real part is x-coordinate, imaginary part is y-coordinate.
What is the geometric effect of adding complex numbers $z_1+z_2$ ?

Vector addition (translate head-to-tail). Place vectors end-to-end to find the sum.
What is the geometric transformation of multiplying by $2i$ ?

Dilation by $2$ and rotation by $90^circ$ . Scale by 2 and rotate 90° counterclockwise.
What complex number corresponds to the point $(4,-9)$ ?

$4-9i$ . Point coordinates become real and imaginary parts.
What is $-i(4-3i)$ ?

$-3-4i$ . Distribute $-i$ and use $i^2 = -1$ .
What is the result of conjugating twice: $bar{(bar{z})}$ ?

$bar{(bar{z})}=z$ . Double conjugation returns the original number.
What is $z-bar{z}$ in terms of $Im(z)$ ?

$z-bar{z}=2Im(z)i$ . Difference gives twice the imaginary part times $i$ .
What is $bar{z_1z_2}$ in terms of $bar{z_1}$ and $bar{z_2}$ ?

$bar{z_1z_2}=bar{z_1}bar{z_2}$ . Conjugate distributes over multiplication.
What is $bar{(z_1+z_2)}$ in terms of conjugates?

$bar{(z_1+z_2)}=bar{z_1}+bar{z_2}$ . Conjugate distributes over addition.
What is $\bar{\frac{z_1}{z_2}}$ for $z_2 \neq 0$ ?

$\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}$ . Conjugate distributes over division.
What happens to arguments under division: $arg(\frac{z_1}{z_2})$ ?

$arg(\frac{z_1}{z_2})=arg(z_1)-arg(z_2)$ . Arguments subtract when complex numbers are divided.
What is the geometric meaning of multiplying by $-i$ ?

Rotation by $90^circ$ clockwise. Multiplication by $-i$ rotates by a quarter turn right.
What is $(1+i)(1-i)$ using conjugates?

$2$ . Difference of squares formula: $(a+bi)(a-bi) = a^2 + b^2$ .
What is the geometric meaning of multiplying by a real $k<0$ ?

Dilation by $k$ and rotation by $180^circ$ . Negative real scaling stretches and reflects through origin.
What is $(2+i)(3-2i)$ ?

$8-i$ . Use FOIL method to expand the product.
What point represents $-z$ if $z=a+bi$ ?

$(-a,-b)$ . Negation reflects the point through the origin.
What is $z \bar{z}$ if $z=3-4i$ ?

$25$ . Apply formula $z \bar{z} = a^2 + b^2$ .
What is $z+bar{z}$ in terms of $Re(z)$ ?

$z+bar{z}=2Re(z)$ . Sum of conjugates equals twice the real part.
What is $zbar{z}$ for $z=a+bi$ ?

$zbar{z}=a^2+b^2$ . Product of a complex number with its conjugate gives modulus squared.
What is the geometric meaning of multiplying by $-1$ ?

Rotation by $180^circ$ about the origin. Multiplication by $-1$ rotates by half a turn.
What is $\bar{(z_1 + z_2)}$ in terms of conjugates?

$\bar{(z_1 + z_2)} = \bar{z_1} + \bar{z_2}$ . Conjugate distributes over addition.
What is the distance between $z_1=1+2i$ and $z_2=4-2i$ ?

$5$ . Distance formula: $|z_1 - z_2| = |(3-4i)| = 5$ .
What is the argument relation between $z$ and $\bar{z}$ (for $z \neq 0$ )?

$\arg(\bar{z}) = -\arg(z)$ . Conjugate has opposite argument from original.
What is the argument relation between $z$ and $bar{z}$ (for $zneq 0$ )?

$arg(bar{z})=-arg(z)$ . Conjugate has opposite argument from original.
What is $(5-2i)-(1+7i)$ ?

$4-9i$ . Subtract real parts and imaginary parts separately.
What is Euler form of a complex number $z$ in polar coordinates?

$z=re^{i\theta}$ . Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$ .
What is $arg(z_1z_2)$ if $arg(z_1)=30^circ$ and $arg(z_2)=70^circ$ ?

$100^circ$ . Arguments add under multiplication.
What is $\Re(z)$ and $\Im(z)$ for $z=a+bi$ ?

$\Re(z)=a$ , $\Im(z)=b$ . Real part is coefficient of 1, imaginary part is coefficient of $i$ .
What is the formula for the reciprocal of $z \neq 0$ using conjugates?

$\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}$ . Multiply by conjugate over modulus squared.
What happens to arguments under division: $\arg(\frac{z_1}{z_2})$ ?

$\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$ . Arguments subtract when complex numbers are divided.
What point on the complex plane represents $z=a+bi$ ?

$(a,b)$ . Real part is the x-coordinate, imaginary part is the y-coordinate.
What is $i(4-3i)$ ?

$3+4i$ . Distribute $i$ and use $i^2 = -1$ .
What is the argument $\theta$ of $z$ on the complex plane?

Angle from positive real axis to $z$ . Measured counterclockwise from positive x-axis.
What is the complex conjugate of $z=a+bi$ ?

$\bar{z} = a - bi$ . Change the sign of the imaginary part only.
What is $zbar{z}$ if $z=3-4i$ ?

$25$ . Apply formula $z\bar{z} = a^2 + b^2$ .
What is the geometric effect of conjugation $z mapsto bar{z}$ ?

Reflection across the real axis. Conjugation mirrors the point across the x-axis.
What is $\frac{1}{2+3i}$ in $a+bi$ form?

$\frac{2}{13}-\frac{3}{13}i$ . Multiply by conjugate and simplify using $z\bar{z} = |z|^2$ .
What is the geometric effect of subtracting $z_1-z_2$ ?

Add $z_1+(-z_2)$ (difference of vectors). Subtract by adding the negative vector.
What happens to arguments under multiplication: $arg(z_1z_2)$ ?

$arg(z_1z_2)=arg(z_1)+arg(z_2)$ . Arguments add when complex numbers are multiplied.
What is $Re(z)$ and $Im(z)$ for $z=a+bi$ ?

$Re(z)=a,Im(z)=b$ . Real part is coefficient of 1, imaginary part is coefficient of $i$ .
What complex number corresponds to the point $(x,y)$ on the complex plane?

$x+yi$ . x-coordinate becomes real part, y-coordinate becomes imaginary part.
What is the geometric meaning of multiplying by a real $k>0$ ?

Dilation by factor $k$ from the origin. Positive real scaling stretches by factor $k$ .

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