HSN.CN Tutor Intake — Complex Numbers

The imaginary unit $i$ is defined so that $i^2 = -1$. Which
statement correctly explains why $i$ is defined this way?

A complex number is written in standard form $a + bi$, where
$a$ and $b$ are real numbers. For the complex number $-5 + 3i$,
which statement is correct?

A student computes the discriminant of $x^2 + 4x + 13 = 0$
and gets $b^2 - 4ac = 16 - 52 = -36$. What does this tell
us about the solutions?

Using the pattern $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$
(cycle of length 4), what is $i^{23}$?

Compute $(3 + 5i) - (7 - 2i)$ and write the result in the form
$a + bi$. Enter the real part $a$ and then the imaginary part $b$
(the coefficient of $i$).

Real part $a =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

Compute $(1 - 6i) + (4 + 2i)$ and write the result in the form
$a + bi$.

Imaginary part $b$ (the coefficient of $i$) $=$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

Compute $(2 + 3i)(1 - 4i)$ and write the result in the form $a + bi$.

Real part $a =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

Compute $(1 + 2i)^2$ and write the result in the form $a + bi$.

Imaginary part $b$ (the coefficient of $i$) $=$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲