Understanding Complex Numbers - Algebra 2
Card 1 of 30
Identify whether $3+0i$ is real, imaginary, or complex with nonzero parts.
Identify whether $3+0i$ is real, imaginary, or complex with nonzero parts.
Tap to reveal answer
Real number. Has zero imaginary part, so it's real.
Real number. Has zero imaginary part, so it's real.
← Didn't Know|Knew It →
What is the real part of the complex number $a+bi$?
What is the real part of the complex number $a+bi$?
Tap to reveal answer
$\operatorname{Re}(a+bi)=a$. The real part is the coefficient of the constant term.
$\operatorname{Re}(a+bi)=a$. The real part is the coefficient of the constant term.
← Didn't Know|Knew It →
What is the imaginary part of $-6+11i$?
What is the imaginary part of $-6+11i$?
Tap to reveal answer
$11$. The imaginary part is the coefficient of $i$.
$11$. The imaginary part is the coefficient of $i$.
← Didn't Know|Knew It →
What is $(-i)^2$ simplified?
What is $(-i)^2$ simplified?
Tap to reveal answer
$-1$. $(-i)^2 = (-1)^2 \cdot i^2 = 1 \cdot (-1) = -1$
$-1$. $(-i)^2 = (-1)^2 \cdot i^2 = 1 \cdot (-1) = -1$
← Didn't Know|Knew It →
What is $\sqrt{-9}$ written using $i$?
What is $\sqrt{-9}$ written using $i$?
Tap to reveal answer
$3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$
$3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$
← Didn't Know|Knew It →
What is $\sqrt{-16}$ written using $i$?
What is $\sqrt{-16}$ written using $i$?
Tap to reveal answer
$4i$. $\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i$
$4i$. $\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i$
← Didn't Know|Knew It →
What is $\sqrt{-25}$ written using $i$?
What is $\sqrt{-25}$ written using $i$?
Tap to reveal answer
$5i$. $\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$
$5i$. $\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$
← Didn't Know|Knew It →
What is $\sqrt{-49}$ written using $i$?
What is $\sqrt{-49}$ written using $i$?
Tap to reveal answer
$7i$. $\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i$
$7i$. $\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i$
← Didn't Know|Knew It →
What is $\sqrt{-64}$ written using $i$?
What is $\sqrt{-64}$ written using $i$?
Tap to reveal answer
$8i$. $\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i$
$8i$. $\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i$
← Didn't Know|Knew It →
What is $\sqrt{-81}$ written using $i$?
What is $\sqrt{-81}$ written using $i$?
Tap to reveal answer
$9i$. $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$
$9i$. $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$
← Didn't Know|Knew It →
What is $\sqrt{-100}$ written using $i$?
What is $\sqrt{-100}$ written using $i$?
Tap to reveal answer
$10i$. $\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i$
$10i$. $\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i$
← Didn't Know|Knew It →
What is $\sqrt{-4}$ written using $i$?
What is $\sqrt{-4}$ written using $i$?
Tap to reveal answer
$2i$. $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$
$2i$. $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$
← Didn't Know|Knew It →
What is $\sqrt{-36}$ written using $i$?
What is $\sqrt{-36}$ written using $i$?
Tap to reveal answer
$6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$
$6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$
← Didn't Know|Knew It →
What is $\sqrt{-121}$ written using $i$?
What is $\sqrt{-121}$ written using $i$?
Tap to reveal answer
$11i$. $\sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i$
$11i$. $\sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i$
← Didn't Know|Knew It →
Which statement is true about $a$ and $b$ in $a+bi$?
Which statement is true about $a$ and $b$ in $a+bi$?
Tap to reveal answer
$a$ and $b$ are real numbers. Both coefficients must be real for complex form.
$a$ and $b$ are real numbers. Both coefficients must be real for complex form.
← Didn't Know|Knew It →
What condition on $b$ makes $a+bi$ a real number?
What condition on $b$ makes $a+bi$ a real number?
Tap to reveal answer
$b=0$. When imaginary part is zero, number is real.
$b=0$. When imaginary part is zero, number is real.
← Didn't Know|Knew It →
What condition on $a$ makes $a+bi$ a pure imaginary number?
What condition on $a$ makes $a+bi$ a pure imaginary number?
Tap to reveal answer
$a=0$. When real part is zero, number is pure imaginary.
$a=0$. When real part is zero, number is pure imaginary.
← Didn't Know|Knew It →
What is $(-i)^3$ simplified?
What is $(-i)^3$ simplified?
Tap to reveal answer
$i$. $(-i)^3 = (-1)^3 \cdot i^3 = -1 \cdot (-i) = i$
$i$. $(-i)^3 = (-1)^3 \cdot i^3 = -1 \cdot (-i) = i$
← Didn't Know|Knew It →
What is $\frac{1}{i}$ simplified in terms of $i$?
What is $\frac{1}{i}$ simplified in terms of $i$?
Tap to reveal answer
$-i$. Multiply by $\frac{-i}{-i}$ to get $\frac{-i}{1} = -i$
$-i$. Multiply by $\frac{-i}{-i}$ to get $\frac{-i}{1} = -i$
← Didn't Know|Knew It →
What are the two real-number components in the complex form $a+bi$?
What are the two real-number components in the complex form $a+bi$?
Tap to reveal answer
$a$ is real part; $b$ is imaginary coefficient. Standard complex form separates real and imaginary components.
$a$ is real part; $b$ is imaginary coefficient. Standard complex form separates real and imaginary components.
← Didn't Know|Knew It →
What is the imaginary unit written as a complex number in $a+bi$ form?
What is the imaginary unit written as a complex number in $a+bi$ form?
Tap to reveal answer
$i=0+1i$. Pure imaginary unit with zero real part.
$i=0+1i$. Pure imaginary unit with zero real part.
← Didn't Know|Knew It →
Identify $a$ and $b$ for the complex number $7-2i$ in the form $a+bi$.
Identify $a$ and $b$ for the complex number $7-2i$ in the form $a+bi$.
Tap to reveal answer
$a=7,\ b=-2$. Real part is 7, imaginary coefficient is -2.
$a=7,\ b=-2$. Real part is 7, imaginary coefficient is -2.
← Didn't Know|Knew It →
Identify $a$ and $b$ for the complex number $-4+9i$ in the form $a+bi$.
Identify $a$ and $b$ for the complex number $-4+9i$ in the form $a+bi$.
Tap to reveal answer
$a=-4,\ b=9$. Real part is -4, imaginary coefficient is 9.
$a=-4,\ b=9$. Real part is -4, imaginary coefficient is 9.
← Didn't Know|Knew It →
What is $i^2$?
What is $i^2$?
Tap to reveal answer
$-1$. By definition of the imaginary unit.
$-1$. By definition of the imaginary unit.
← Didn't Know|Knew It →
What is $i^4$ simplified?
What is $i^4$ simplified?
Tap to reveal answer
$1$. $i^4 = (i^2)^2 = (-1)^2 = 1$
$1$. $i^4 = (i^2)^2 = (-1)^2 = 1$
← Didn't Know|Knew It →
What is $i^7$ simplified?
What is $i^7$ simplified?
Tap to reveal answer
$-i$. $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$
$-i$. $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$
← Didn't Know|Knew It →
What is $(-i)^3$ simplified?
What is $(-i)^3$ simplified?
Tap to reveal answer
$i$. $(-i)^3 = (-1)^3 \cdot i^3 = -1 \cdot (-i) = i$
$i$. $(-i)^3 = (-1)^3 \cdot i^3 = -1 \cdot (-i) = i$
← Didn't Know|Knew It →
What is $i^3$ simplified using $i^2=-1$?
What is $i^3$ simplified using $i^2=-1$?
Tap to reveal answer
$-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$
$-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$
← Didn't Know|Knew It →
What is $\sqrt{-81}$ written using $i$?
What is $\sqrt{-81}$ written using $i$?
Tap to reveal answer
$9i$. $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$
$9i$. $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$
← Didn't Know|Knew It →
What is $\sqrt{-121}$ written using $i$?
What is $\sqrt{-121}$ written using $i$?
Tap to reveal answer
$11i$. $\sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i$
$11i$. $\sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i$
← Didn't Know|Knew It →