Complex Numbers - ACT Math
Card 1 of 30
What is the result of $i^3$?
What is the result of $i^3$?
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-i. Since $i^3 = i^2 \cdot i = (-1) \cdot i = -i$.
-i. Since $i^3 = i^2 \cdot i = (-1) \cdot i = -i$.
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What is $(3+2i)+(4-5i)$ in standard form?
What is $(3+2i)+(4-5i)$ in standard form?
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$7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.
$7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.
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What is $(3+2i)+(4-5i)$ in standard form?
What is $(3+2i)+(4-5i)$ in standard form?
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$7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.
$7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.
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State the polar form of the complex number $1 + 0i$.
State the polar form of the complex number $1 + 0i$.
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$1(\cos(0) + i\sin(0))$. Real number 1 has modulus 1 and argument 0.
$1(\cos(0) + i\sin(0))$. Real number 1 has modulus 1 and argument 0.
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What is $i^4$ equal to?
What is $i^4$ equal to?
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$1$. $i^4 = (i^2)^2 = (-1)^2 = 1$.
$1$. $i^4 = (i^2)^2 = (-1)^2 = 1$.
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What is $i^2$ equal to?
What is $i^2$ equal to?
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-1. By definition, $i^2 = -1$.
-1. By definition, $i^2 = -1$.
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What is $\overline{\frac{z_1}{z_2}}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?
What is $\overline{\frac{z_1}{z_2}}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?
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$\frac{\overline{z_1}}{\overline{z_2}}$. Conjugate of quotient equals quotient of conjugates.
$\frac{\overline{z_1}}{\overline{z_2}}$. Conjugate of quotient equals quotient of conjugates.
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What is the rule for adding complex numbers $(a+bi)+(c+di)$?
What is the rule for adding complex numbers $(a+bi)+(c+di)$?
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$(a+c)+(b+d)i$. Add real parts and imaginary parts separately.
$(a+c)+(b+d)i$. Add real parts and imaginary parts separately.
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What is the rule for subtracting complex numbers $(a+bi)-(c+di)$?
What is the rule for subtracting complex numbers $(a+bi)-(c+di)$?
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$(a-c)+(b-d)i$. Subtract real parts and imaginary parts separately.
$(a-c)+(b-d)i$. Subtract real parts and imaginary parts separately.
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What is the product $(a+bi)(c+di)$ in standard form?
What is the product $(a+bi)(c+di)$ in standard form?
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$(ac-bd)+(ad+bc)i$. Use FOIL: $(a+bi)(c+di) = ac + adi + bci + bdi^2$.
$(ac-bd)+(ad+bc)i$. Use FOIL: $(a+bi)(c+di) = ac + adi + bci + bdi^2$.
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What is $\frac{1}{a+bi}$ in standard form (with $a,b\in\mathbb{R}$)?
What is $\frac{1}{a+bi}$ in standard form (with $a,b\in\mathbb{R}$)?
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$\frac{a-bi}{a^2+b^2}$. Multiply by conjugate: $\frac{1}{a+bi} \cdot \frac{a-bi}{a-bi}$.
$\frac{a-bi}{a^2+b^2}$. Multiply by conjugate: $\frac{1}{a+bi} \cdot \frac{a-bi}{a-bi}$.
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What is the condition for two complex numbers $a+bi$ and $c+di$ to be equal?
What is the condition for two complex numbers $a+bi$ and $c+di$ to be equal?
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$a=c$ and $b=d$. Real and imaginary parts must match separately.
$a=c$ and $b=d$. Real and imaginary parts must match separately.
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What is $\sqrt{-9}$ written using $i$?
What is $\sqrt{-9}$ written using $i$?
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$3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.
$3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.
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What is $\sqrt{-50}$ written in simplest form using $i$?
What is $\sqrt{-50}$ written in simplest form using $i$?
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$5\sqrt{2},i$. $\sqrt{-50} = \sqrt{25 \cdot 2} \cdot i = 5\sqrt{2}i$.
$5\sqrt{2},i$. $\sqrt{-50} = \sqrt{25 \cdot 2} \cdot i = 5\sqrt{2}i$.
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What is $i^{17}$ equal to?
What is $i^{17}$ equal to?
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$i$. $17 = 4 \cdot 4 + 1$, so $i^{17} = i^1 = i$.
$i$. $17 = 4 \cdot 4 + 1$, so $i^{17} = i^1 = i$.
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What is $i^{22}$ equal to?
What is $i^{22}$ equal to?
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$-1$. $22 = 4 \cdot 5 + 2$, so $i^{22} = i^2 = -1$.
$-1$. $22 = 4 \cdot 5 + 2$, so $i^{22} = i^2 = -1$.
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What is $(6-7i)-(2+3i)$ in standard form?
What is $(6-7i)-(2+3i)$ in standard form?
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$4-10i$. Subtract real: $6-2=4$, imaginary: $-7-3=-10$.
$4-10i$. Subtract real: $6-2=4$, imaginary: $-7-3=-10$.
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Express $i^5$ in terms of $i$.
Express $i^5$ in terms of $i$.
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$i$. Use $i^5 = i^4 \cdot i = 1 \cdot i = i$.
$i$. Use $i^5 = i^4 \cdot i = 1 \cdot i = i$.
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What is $(2+i)(3-4i)$ in standard form?
What is $(2+i)(3-4i)$ in standard form?
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$10-5i$. $(2+i)(3-4i) = 6-8i+3i-4i^2 = 6-5i+4 = 10-5i$.
$10-5i$. $(2+i)(3-4i) = 6-8i+3i-4i^2 = 6-5i+4 = 10-5i$.
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What is $i^7$ in terms of $i$?
What is $i^7$ in terms of $i$?
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$-i$. Use $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$.
$-i$. Use $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$.
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What does the imaginary unit $i$ represent?
What does the imaginary unit $i$ represent?
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The square root of -1. The imaginary unit $i$ is defined as $\sqrt{-1}$.
The square root of -1. The imaginary unit $i$ is defined as $\sqrt{-1}$.
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What is the sum of $3 + 4i$ and $1 + 2i$?
What is the sum of $3 + 4i$ and $1 + 2i$?
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$4 + 6i$. Add real parts and imaginary parts separately: $(3+1) + (4+2)i$.
$4 + 6i$. Add real parts and imaginary parts separately: $(3+1) + (4+2)i$.
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Identify the real part of the complex number $7 + 3i$.
Identify the real part of the complex number $7 + 3i$.
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- The real part is the term without $i$.
- The real part is the term without $i$.
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State the value of $i^4$.
State the value of $i^4$.
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- Since $i^4 = (i^2)^2 = (-1)^2 = 1$.
- Since $i^4 = (i^2)^2 = (-1)^2 = 1$.
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Identify the modulus of $0 + 7i$.
Identify the modulus of $0 + 7i$.
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- For pure imaginary $bi$, modulus is $|b|$.
- For pure imaginary $bi$, modulus is $|b|$.
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Find the real component of $(2 + 3i)(4 + 5i)$.
Find the real component of $(2 + 3i)(4 + 5i)$.
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-7. Expand: $8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$.
-7. Expand: $8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$.
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What is the inverse of the complex number $1 + i$ in standard form?
What is the inverse of the complex number $1 + i$ in standard form?
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$\frac{1}{2} - \frac{1}{2}i$. Multiply by conjugate: $\frac{1+i}{(1+i)(1-i)} = \frac{1-i}{2}$.
$\frac{1}{2} - \frac{1}{2}i$. Multiply by conjugate: $\frac{1+i}{(1+i)(1-i)} = \frac{1-i}{2}$.
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Calculate the product of $i$ and $4i$.
Calculate the product of $i$ and $4i$.
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-4. Since $i \cdot 4i = 4i^2 = 4(-1) = -4$.
-4. Since $i \cdot 4i = 4i^2 = 4(-1) = -4$.
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What is the square of the complex number $1 + i$?
What is the square of the complex number $1 + i$?
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2i. $(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$.
2i. $(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$.
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If $z = 4 - 3i$, what is $z^*$?
If $z = 4 - 3i$, what is $z^*$?
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$4 + 3i$. The conjugate of $4 - 3i$ is $4 + 3i$.
$4 + 3i$. The conjugate of $4 - 3i$ is $4 + 3i$.
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