Geometric Representations of Complex Numbers - Algebra 2
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What is $z-bar{z}$ in terms of $Im(z)$?
What is $z-bar{z}$ in terms of $Im(z)$?
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$z-bar{z}=2Im(z)i$. Difference gives twice the imaginary part times $i$.
$z-bar{z}=2Im(z)i$. Difference gives twice the imaginary part times $i$.
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What is the geometric meaning of multiplying by $-1$?
What is the geometric meaning of multiplying by $-1$?
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Rotation by $180^circ$ about the origin. Multiplication by $-1$ rotates by half a turn.
Rotation by $180^circ$ about the origin. Multiplication by $-1$ rotates by half a turn.
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What is the geometric meaning of multiplying by $i$?
What is the geometric meaning of multiplying by $i$?
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Rotation by $90^circ$ counterclockwise. Multiplication by $i$ rotates by a quarter turn left.
Rotation by $90^circ$ counterclockwise. Multiplication by $i$ rotates by a quarter turn left.
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What is $z+bar{z}$ in terms of $Re(z)$?
What is $z+bar{z}$ in terms of $Re(z)$?
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$z+bar{z}=2Re(z)$. Sum of conjugates equals twice the real part.
$z+bar{z}=2Re(z)$. Sum of conjugates equals twice the real part.
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What is the geometric meaning of multiplying by a real $k>0$?
What is the geometric meaning of multiplying by a real $k>0$?
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Dilation by factor $k$ from the origin. Positive real scaling stretches by factor $k$.
Dilation by factor $k$ from the origin. Positive real scaling stretches by factor $k$.
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What is $\bar{\frac{z_1}{z_2}}$ for $z_2 \neq 0$?
What is $\bar{\frac{z_1}{z_2}}$ for $z_2 \neq 0$?
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$\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}$. Conjugate distributes over division.
$\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}$. Conjugate distributes over division.
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What is $i(4-3i)$?
What is $i(4-3i)$?
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$3+4i$. Distribute $i$ and use $i^2 = -1$.
$3+4i$. Distribute $i$ and use $i^2 = -1$.
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What is $zbar{z}$ for $z=a+bi$?
What is $zbar{z}$ for $z=a+bi$?
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$zbar{z}=a^2+b^2$. Product of a complex number with its conjugate gives modulus squared.
$zbar{z}=a^2+b^2$. Product of a complex number with its conjugate gives modulus squared.
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What is the geometric meaning of multiplying by $-i$?
What is the geometric meaning of multiplying by $-i$?
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Rotation by $90^circ$ clockwise. Multiplication by $-i$ rotates by a quarter turn right.
Rotation by $90^circ$ clockwise. Multiplication by $-i$ rotates by a quarter turn right.
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What is the geometric meaning of multiplying by a real $k<0$?
What is the geometric meaning of multiplying by a real $k<0$?
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Dilation by $k$ and rotation by $180^circ$. Negative real scaling stretches and reflects through origin.
Dilation by $k$ and rotation by $180^circ$. Negative real scaling stretches and reflects through origin.
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What happens to arguments under multiplication: $arg(z_1z_2)$?
What happens to arguments under multiplication: $arg(z_1z_2)$?
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$arg(z_1z_2)=arg(z_1)+arg(z_2)$. Arguments add when complex numbers are multiplied.
$arg(z_1z_2)=arg(z_1)+arg(z_2)$. Arguments add when complex numbers are multiplied.
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What is $bar{z_1z_2}$ in terms of $bar{z_1}$ and $bar{z_2}$?
What is $bar{z_1z_2}$ in terms of $bar{z_1}$ and $bar{z_2}$?
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$bar{z_1z_2}=bar{z_1}bar{z_2}$. Conjugate distributes over multiplication.
$bar{z_1z_2}=bar{z_1}bar{z_2}$. Conjugate distributes over multiplication.
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What is $bar{(z_1+z_2)}$ in terms of conjugates?
What is $bar{(z_1+z_2)}$ in terms of conjugates?
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$bar{(z_1+z_2)}=bar{z_1}+bar{z_2}$. Conjugate distributes over addition.
$bar{(z_1+z_2)}=bar{z_1}+bar{z_2}$. Conjugate distributes over addition.
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What is the result of conjugating twice: $bar{(bar{z})}$?
What is the result of conjugating twice: $bar{(bar{z})}$?
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$bar{(bar{z})}=z$. Double conjugation returns the original number.
$bar{(bar{z})}=z$. Double conjugation returns the original number.
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What is the argument relation between $z$ and $bar{z}$ (for $zneq 0$)?
What is the argument relation between $z$ and $bar{z}$ (for $zneq 0$)?
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$arg(bar{z})=-arg(z)$. Conjugate has opposite argument from original.
$arg(bar{z})=-arg(z)$. Conjugate has opposite argument from original.
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What is $Re(z)$ and $Im(z)$ for $z=a+bi$?
What is $Re(z)$ and $Im(z)$ for $z=a+bi$?
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$Re(z)=a,Im(z)=b$. Real part is coefficient of 1, imaginary part is coefficient of $i$.
$Re(z)=a,Im(z)=b$. Real part is coefficient of 1, imaginary part is coefficient of $i$.
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What is the geometric meaning of $z_1z_2$ in polar form?
What is the geometric meaning of $z_1z_2$ in polar form?
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Multiply moduli; add arguments (scale and rotate). Multiplication scales and rotates simultaneously.
Multiply moduli; add arguments (scale and rotate). Multiplication scales and rotates simultaneously.
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What is $(5-2i)-(1+7i)$?
What is $(5-2i)-(1+7i)$?
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$4-9i$. Subtract real parts and imaginary parts separately.
$4-9i$. Subtract real parts and imaginary parts separately.
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What happens to arguments under division: $\arg(\frac{z_1}{z_2})$?
What happens to arguments under division: $\arg(\frac{z_1}{z_2})$?
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$\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$. Arguments subtract when complex numbers are divided.
$\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$. Arguments subtract when complex numbers are divided.
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What is the complex conjugate of $z=a+bi$?
What is the complex conjugate of $z=a+bi$?
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$\bar{z} = a - bi$. Change the sign of the imaginary part only.
$\bar{z} = a - bi$. Change the sign of the imaginary part only.
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What is $z \bar{z}$ if $z=3-4i$?
What is $z \bar{z}$ if $z=3-4i$?
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$25$. Apply formula $z \bar{z} = a^2 + b^2$.
$25$. Apply formula $z \bar{z} = a^2 + b^2$.
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What is the conjugate of $-6+11i$?
What is the conjugate of $-6+11i$?
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$-6-11i$. Negate the imaginary part to find conjugate.
$-6-11i$. Negate the imaginary part to find conjugate.
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What point on the complex plane represents $z=a+bi$?
What point on the complex plane represents $z=a+bi$?
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$(a,b)$. Real part is the x-coordinate, imaginary part is the y-coordinate.
$(a,b)$. Real part is the x-coordinate, imaginary part is the y-coordinate.
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What complex number corresponds to the point $(x,y)$ on the complex plane?
What complex number corresponds to the point $(x,y)$ on the complex plane?
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$x+yi$. x-coordinate becomes real part, y-coordinate becomes imaginary part.
$x+yi$. x-coordinate becomes real part, y-coordinate becomes imaginary part.
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What is the argument $ heta$ of $z$ on the complex plane?
What is the argument $ heta$ of $z$ on the complex plane?
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Angle from positive real axis to $z$. Measured counterclockwise from positive x-axis.
Angle from positive real axis to $z$. Measured counterclockwise from positive x-axis.
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What is Euler form of a complex number $z$ in polar coordinates?
What is Euler form of a complex number $z$ in polar coordinates?
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$z=re^{i heta}$. Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$.
$z=re^{i heta}$. Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$.
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What is the geometric effect of adding complex numbers $z_1+z_2$?
What is the geometric effect of adding complex numbers $z_1+z_2$?
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Vector addition (translate head-to-tail). Place vectors end-to-end to find the sum.
Vector addition (translate head-to-tail). Place vectors end-to-end to find the sum.
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What is the geometric effect of subtracting $z_1-z_2$?
What is the geometric effect of subtracting $z_1-z_2$?
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Add $z_1+(-z_2)$ (difference of vectors). Subtract by adding the negative vector.
Add $z_1+(-z_2)$ (difference of vectors). Subtract by adding the negative vector.
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What point represents $-z$ if $z=a+bi$?
What point represents $-z$ if $z=a+bi$?
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$(-a,-b)$. Negation reflects the point through the origin.
$(-a,-b)$. Negation reflects the point through the origin.
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What is the geometric effect of conjugation $z mapsto bar{z}$?
What is the geometric effect of conjugation $z mapsto bar{z}$?
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Reflection across the real axis. Conjugation mirrors the point across the x-axis.
Reflection across the real axis. Conjugation mirrors the point across the x-axis.
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