Algebra 2 Flashcards: Geometric Representations Of Complex Numbers

What this deck covers

This deck focuses on Geometric Representations Of Complex Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $\bar{(\frac{z_1}{z_2})}$ for $z_2 \neq 0$?

Answer: $\bar{(\frac{z_1}{z_2})} = \frac{\bar{z_1}}{\bar{z_2}}$. Conjugate distributes over division.

Flashcard 2: What is the geometric meaning of multiplying by $i$?

Answer: Rotation by 90^circ counterclockwise. Multiplication by $i$ rotates by a quarter turn left.

Flashcard 3: What is the geometric meaning of $z_1z_2$ in polar form?

Answer: Multiply moduli; add arguments (scale and rotate). Multiplication scales and rotates simultaneously.

Flashcard 4: What is the conjugate of $-6+11i$?

Answer: $-6-11i$. Negate the imaginary part to find conjugate.

Flashcard 5: What is the point for $z=-2+7i$ on the complex plane?

Answer: $(-2,7)$. Real part is x-coordinate, imaginary part is y-coordinate.

Flashcard 6: What is the geometric effect of adding complex numbers $z_1+z_2$?

Answer: Vector addition (translate head-to-tail). Place vectors end-to-end to find the sum.

Flashcard 7: What is the geometric transformation of multiplying by $2i$?

Answer: Dilation by $2$ and rotation by 90^circ. Scale by 2 and rotate 90° counterclockwise.

Flashcard 8: What complex number corresponds to the point $(4,-9)$?

Answer: $4-9i$. Point coordinates become real and imaginary parts.

Flashcard 9: What is $-i(4-3i)$?

Answer: $-3-4i$. Distribute $-i$ and use $i^2 = -1$.

Flashcard 10: What is the result of conjugating twice: bar{(bar{z})}?

Answer: bar{(bar{z})}=z. Double conjugation returns the original number.

Flashcard 11: What is z-bar{z} in terms of Im(z)?

Answer: z-bar{z}=2Im(z)i. Difference gives twice the imaginary part times $i$.

Flashcard 12: What is bar{z_1z_2} in terms of bar{z_1} and bar{z_2}?

Answer: bar{z_1z_2}=bar{z_1}bar{z_2}. Conjugate distributes over multiplication.

Flashcard 13: What is bar{(z_1+z_2)} in terms of conjugates?

Answer: bar{(z_1+z_2)}=bar{z_1}+bar{z_2}. Conjugate distributes over addition.

Flashcard 14: What is $\bar{\frac{z_1}{z_2}}$ for $z_2 \neq 0$?

Answer: $\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}$. Conjugate distributes over division.

Flashcard 15: What happens to arguments under division: arg(rac{z_1}{z_2})?

Answer: arg(rac{z_1}{z_2})=arg(z_1)-arg(z_2). Arguments subtract when complex numbers are divided.

Flashcard 16: What is the geometric meaning of multiplying by $-i$?

Answer: Rotation by 90^circ clockwise. Multiplication by $-i$ rotates by a quarter turn right.

Flashcard 17: What is $(1+i)(1-i)$ using conjugates?

Answer: $2$. Difference of squares formula: $(a+bi)(a-bi) = a^2 + b^2$.

Flashcard 18: What is the geometric meaning of multiplying by a real $k<0$?

Answer: Dilation by k and rotation by 180^circ. Negative real scaling stretches and reflects through origin.

Flashcard 19: What is $(2+i)(3-2i)$?

Answer: $8-i$. Use FOIL method to expand the product.

Flashcard 20: What point represents $-z$ if $z=a+bi$?

Answer: $(-a,-b)$. Negation reflects the point through the origin.

Flashcard 21: What is $z \bar{z}$ if $z=3-4i$?

Answer: $25$. Apply formula $z \bar{z} = a^2 + b^2$.

Flashcard 22: What is z+bar{z} in terms of Re(z)?

Answer: z+bar{z}=2Re(z). Sum of conjugates equals twice the real part.

Flashcard 23: What is zbar{z} for $z=a+bi$?

Answer: zbar{z}=a^2+b^2. Product of a complex number with its conjugate gives modulus squared.

Flashcard 24: What is the geometric meaning of multiplying by $-1$?

Answer: Rotation by 180^circ about the origin. Multiplication by $-1$ rotates by half a turn.

Flashcard 25: What is $\bar{(z_1 + z_2)}$ in terms of conjugates?

Answer: $\bar{(z_1 + z_2)} = \bar{z_1} + \bar{z_2}$. Conjugate distributes over addition.

Flashcard 26: What is the distance between $z_1=1+2i$ and $z_2=4-2i$?

Answer: $5$. Distance formula: $|z_1 - z_2| = |(3-4i)| = 5$.

Flashcard 27: What is the argument relation between $z$ and $\bar{z}$ (for $z \neq 0$)?

Answer: $\arg(\bar{z}) = -\arg(z)$. Conjugate has opposite argument from original.

Flashcard 28: What is the argument relation between $z$ and bar{z} (for zneq 0)?

Answer: arg(bar{z})=-arg(z). Conjugate has opposite argument from original.

Flashcard 29: What is (5-2i)-(1+7i)?

Answer: $4-9i$. Subtract real parts and imaginary parts separately.

Flashcard 30: What is Euler form of a complex number $z$ in polar coordinates?

Answer: $z=re^{i heta}$. Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$.

Flashcard 31: What is arg(z_1z_2) if arg(z_1)=30^circ and arg(z_2)=70^circ?

Answer: 100^circ. Arguments add under multiplication.

Flashcard 32: What is $\Re(z)$ and $\Im(z)$ for $z=a+bi$?

Answer: $\Re(z)=a$, $\Im(z)=b$. Real part is coefficient of 1, imaginary part is coefficient of $i$.

Flashcard 33: What is the formula for the reciprocal of $z \neq 0$ using conjugates?

Answer: $\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}$. Multiply by conjugate over modulus squared.

Flashcard 34: What happens to arguments under division: $\arg(\frac{z_1}{z_2})$?

Answer: $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$. Arguments subtract when complex numbers are divided.

Flashcard 35: What point on the complex plane represents $z=a+bi$?

Answer: $(a,b)$. Real part is the x-coordinate, imaginary part is the y-coordinate.

Flashcard 36: What is $i(4-3i)$?

Answer: $3+4i$. Distribute $i$ and use $i^2 = -1$.

Flashcard 37: What is the argument  heta of $z$ on the complex plane?

Answer: Angle from positive real axis to $z$. Measured counterclockwise from positive x-axis.

Flashcard 38: What is the complex conjugate of $z=a+bi$?

Answer: $\bar{z} = a - bi$. Change the sign of the imaginary part only.

Flashcard 39: What is zbar{z} if $z=3-4i$?

Answer: $25$. Apply formula $z\bar{z} = a^2 + b^2$.

Flashcard 40: What is the geometric effect of conjugation z mapsto bar{z}?

Answer: Reflection across the real axis. Conjugation mirrors the point across the x-axis.

Flashcard 41: What is rac{1}{2+3i} in $a+bi$ form?

Answer: rac{2}{13}-rac{3}{13}i. Multiply by conjugate and simplify using $z\bar{z} = |z|^2$.

Flashcard 42: What is the geometric effect of subtracting $z_1-z_2$?

Answer: Add $z_1+(-z_2)$ (difference of vectors). Subtract by adding the negative vector.

Flashcard 43: What happens to arguments under multiplication: arg(z_1z_2)?

Answer: arg(z_1z_2)=arg(z_1)+arg(z_2). Arguments add when complex numbers are multiplied.

Flashcard 44: What is Re(z) and Im(z) for $z=a+bi$?

Answer: Re(z)=a,Im(z)=b. Real part is coefficient of 1, imaginary part is coefficient of $i$.

Flashcard 45: What complex number corresponds to the point $(x,y)$ on the complex plane?

Answer: $x+yi$. x-coordinate becomes real part, y-coordinate becomes imaginary part.

Flashcard 46: What is the geometric meaning of multiplying by a real $k>0$?

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