Math, Complex Numbers and the Imaginary Unit Worksheet

Question 1

Define the imaginary unit i and explain what i squared equals.

Accepted Answer

The imaginary unit i is defined as the square root of negative 1. This means that i squared equals negative 1.

Question 2

Simplify the expression square root of negative 49.

Accepted Answer

The expression simplifies to 7i because square root of negative 49 equals square root of 49 times square root of negative 1, which is 7i.

Question 3

Simplify i to the 3rd power.

Accepted Answer

i to the 3rd power equals negative i because i squared is negative 1, and negative 1 times i is negative i.

Question 4

Simplify i to the 4th power.

Accepted Answer

i to the 4th power equals 1 because i squared is negative 1, and negative 1 squared is 1.

Question 5

Simplify i to the 27th power.

Accepted Answer

i to the 27th power equals negative i. The powers of i repeat every 4, and 27 divided by 4 leaves a remainder of 3, so this matches i to the 3rd power, which is negative i.

Question 6

Write the complex number 6 - 2i in the form a + bi and identify its real and imaginary parts.

Accepted Answer

The number 6 - 2i is already in the form a + bi. Its real part is 6, and its imaginary part is negative 2.

Question 7

Add the complex numbers (4 + 3i) + (2 - 5i).

Accepted Answer

The sum is 6 - 2i because the real parts add to 6 and the imaginary parts add to negative 2i.

Question 8

Subtract the complex numbers (7 - 4i) - (3 + 6i).

Accepted Answer

The difference is 4 - 10i because 7 minus 3 equals 4 and negative 4i minus 6i equals negative 10i.

Question 9

Multiply the complex numbers (3i)(-4i).

Accepted Answer

The product is 12 because 3 times negative 4 is negative 12 and i squared is negative 1, so negative 12 times negative 1 equals 12.

Question 10

Multiply the binomials (2 + 3i)(1 - i).

Accepted Answer

The product is 5 + i. Expanding gives 2 - 2i + 3i - 3i squared, and since i squared is negative 1, this becomes 2 + i + 3, which simplifies to 5 + i.

Question 11

Simplify the expression (5 + 9i) + (-2 - 4i).

Accepted Answer

The expression simplifies to 3 + 5i because 5 plus negative 2 is 3 and 9i plus negative 4i is 5i.

Question 12

Find the complex conjugate of 8 - 11i.

Accepted Answer

The complex conjugate of 8 - 11i is 8 + 11i because the sign of the imaginary term changes.

Question 13

Express the square root of negative 12 in simplest imaginary form.

Accepted Answer

The square root of negative 12 simplifies to 2i square root of 3. This is because square root of negative 12 equals square root of 4 times 3 times square root of negative 1, which becomes 2i square root of 3.

Question 14

Solve the equation x squared + 9 = 0 over the complex numbers.

Accepted Answer

The solutions are x equals 3i and x equals negative 3i. This is because x squared equals negative 9, so x equals plus or minus the square root of negative 9, which is plus or minus 3i.

Question 15

Classify the number -5 + 0i as real, imaginary, or complex, and explain your answer.

Accepted Answer

The number negative 5 + 0i is a real number. It is also a complex number, but it is classified as real because its imaginary part is 0.

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Math, Complex Numbers and the Imaginary Unit

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