Understanding Complex Numbers: Real Part, Imaginary Unit, and Imaginary Part

Equation	Unsolvable in	Solvable in
$x + 2 = 1$	Natural numbers $N$	Integer numbers $Z$
$2 x - 3 = 0$	Integer numbers $Z$	Rational numbers $Q$
$x^{2} = 5$	Rational numbers $Q$	Irrational numbers
$x^{2} = - 2$	Real numbers $R$	$?$

Equation

Unsolvable in

Solvable in

x + 2 = 1

Natural numbers

N

Integer numbers

Z

2 x - 3 = 0

Integer numbers

Z

Rational numbers

Q

x^{2} = 5

Rational numbers

Q

Irrational numbers

x^{2} = - 2

Real numbers

R

?

	Square	Triangle
Area Formula	$A_{□} = s^{2}$	$A_{△} = \frac{1}{2} b h$
Substitution	$A_{□} = x^{2}$	$A_{△} = \frac{1}{2} \cdot 3 \cdot 4 = 6$

Square

Triangle

Area Formula

A_{□} = s^{2}

A_{△} = \frac{1}{2} b h

Substitution

A_{□} = x^{2}

A_{△} = \frac{1}{2} \cdot 3 \cdot 4 = 6

	Square	Triangle
Area	$A_{□} = x^{2}$	$A_{△} = 6$

Square

Triangle

Area

A_{□} = x^{2}

A_{△} = 6

Finding $i^{n}$
If the remainder of $\frac{n}{4}$ is	Then, $i^{n}$ is equal to
$0$	$i^{0} = 1$
$1$	$i^{1} = i$
$2$	$i^{2} = - 1$
$3$	$i^{3} = - i$

Finding

i^{n}

If the remainder of

\frac{n}{4}

Then,

i^{n}

is equal to

0

i^{0} = 1

1

i^{1} = i

2

i^{2} = - 1

3

i^{3} = - i

Component	Impedance
Resistor	$8 Ω$
Capacitor	$- 6 i Ω$
Inductor	$10 i Ω$

Component

Impedance

Resistor

8 Ω

Capacitor

- 6 i Ω

Inductor

10 i Ω

Component	Resistance or Reactance	Impedance
Resistor $1$	$6 Ω$	$6 Ω$
Capacitor	$9 Ω$	$- 9 i Ω$
Inductor	$4 Ω$	$4 i Ω$
Resistor $2$	$3 Ω$	$3 Ω$

Component

Resistance or Reactance

Impedance

Resistor

1

6 Ω

6 Ω

Capacitor

9 Ω

- 9 i Ω

Inductor

4 Ω

4 i Ω

Resistor

2

3 Ω

3 Ω

Simplify each number by using the imaginary unit $i .$

- 49

- 50

- 5 - 11

We want to simplify the given numeric expression by using the imaginary unit i. To do so, we will first recall that the imaginary unit i is a complex number whose square is - 1. i^2=- 1 and i=sqrt(- 1) We can simplify the given expression by using this definition.

We will now simplify sqrt(- 50) by using the fact that i=sqrt(- 1).

In order to write the given expression in its simplest form, we will also simplify sqrt(50) by breaking 50 into two smaller numbers, one of which is a perfect square — namely, 25 and 2.

Finally, we will simplify - 5sqrt(- 11). Note that 11 is a prime number and its only positive factors are 1 and 11.

Which number does not belong with the other three?

A . B . C . D . 2 + - 9 0 + 5 i 5 - 2 i 3 - 8

A complex number written in standard form is a number a+bi, where a and b are real numbers. The number a is the real part and the number b is the imaginary part. If b≠ 0, then a+bi is an imaginary number. If b=0, then a+bi is a real number. a+ bi With this information in mind, Let's carefully examine the given numbers. First, we should ensure that each number is in standard form. We can then identify the real and imaginary parts of the complex numbers.

Option A

We will simplify the number 2+sqrt(- 9) and write it in standard form. To do so, recall that the imaginary unit i is equal to sqrt(- 1).

The number 2 is the real part of this complex number, and the number 3 is the imaginary part.

Option B

The complex number 0+ 5i is already in standard form, so 0 is the real part and 5 is the imaginary part.

Option C

Let's rewrite the number sqrt(5)-2i to identify the real and imaginary parts. sqrt(5)-2i = sqrt(5)+( - 2)i Here, sqrt(5) is the real part and - 2 is the imaginary part.

Option D

We will simplify 3-sqrt(8).

Let's write this number in the form of a+ bi. 3-2sqrt(2) = ( 3-2sqrt(2))+ 0i The real part is 3-2sqrt(2) and the imaginary part is 0.

Conclusion

Finally, we will identify the number that does not belong with the other three.

Option	Number	Standard Form a+bi	Real Part a	Imaginary Part b	Type of Number
A	2+sqrt(- 9)	sqrt(2)+ 3i	sqrt(2)	3	Imaginary Number
B	0+5i	0+ 5i	0	5	Imaginary Number
C	sqrt(5)-2i	sqrt(5)+( - 2)i	sqrt(5)	- 2	Imaginary Number
D	3-sqrt(8)	( 3-2sqrt(2))+ 0i	3-2sqrt(2)	0	Real Number

Looking at the last column, we can see that 3-sqrt(8) is a real number, while the other three are all imaginary numbers. Therefore, 3-sqrt(8) does not belong with the other three. The answer is option D.

Pair the numbers with their correct description.

A complex number written in standard form is a number a+bi, where a and b are real numbers. The number a is the real part and the number b is the imaginary part. a+ bi Depending on the values of a and b, there are three possible cases.

If b≠ 0, then a+bi is an imaginary number.
If b=0, then a+bi is a real number.
If a=0 and b≠ 0, then a+bi is a pure imaginary number.

With this information, we can examine and classify the given numbers.

Number	Real Part a	Imaginary Part b	Type of Number
5+sqrt(7)	5+sqrt(7)	0	Real Number
1+12i	1	12	Imaginary Number
4i	0	4	Pure Imaginary Number

Determine the real and imaginary parts of the complex number

3 - i 2 .

A . B . C . D . Real Part : 3 Imaginary Part : i 2 Real Part : 3 Imaginary Part : - i 2 Real Part : 3 Imaginary Part : 2 Real Part : 3 Imaginary Part : - 2

Determine the real and imaginary parts of the complex number

7 i - 3 .

A . B . C . D . Real Part : 7 Imaginary Part : 3 Real Part : 7 Imaginary Part : - 3 Real Part : - 3 Imaginary Part : 7 Real Part : - 3 Imaginary Part : 7 i

A complex number written in standard form is a number a+bi, where a and b are real numbers. The number a is the real part and the number b is the imaginary part. a+ bi With this information, we can determine the real and imaginary parts of the given number. 3-isqrt(2) = 3+( - sqrt(2))i The real part of the complex number is 3 and the imaginary part is - sqrt(2). Therefore, the answer is option D.

We will now determine the real and imaginary parts of 7i-sqrt(3). To do so, let's first rewrite the complex number by using the Commutative Property of Addition. 7i-sqrt(3)= -sqrt(3)+ 7i Recall that the number a is the real part and the number b is the imaginary part of a complex number written in standard form a+ bi, where a and b are real numbers. This means that the real part of the given complex number is - sqrt(3) and the imaginary part is 7. The answer is option C.

Add or subtract the numbers. Write the answer in standard form.

(3 + 7 i) + (7 - 9 i)

5 - 3 i - (12 - 15 i)

99 - 2 (47 - 48 i) + 4 i - 95 i

To add or subtract two or more complex numbers, we combine their like terms — real parts with real parts and imaginary parts with imaginary parts. In this case, we will remove the parentheses first.

Again, we need to combine like terms — the real parts and imaginary parts. To do so, we will first remove the parenthesis by distributing -1 to the numbers in the parentheses.

Finally, we will simplify the given expression. To do so, we will start by distributing -2 to the numbers in the parentheses, and then we will combine like terms.

Find the values of

x

and

y

that make the following equation true.

9 + (3 y - 7) i = 7 + x - 3 i

We will start by identifying the real parts and imaginary parts of the complex numbers on each side of the given equation. 9+(3y-7)i&=7+x-3i &⇕ 9+( 3y-7)i&= 7+x+( -3)i To find the values of x and y that make this equation true, we need to set the real parts and the imaginary parts of each complex number equal to each other. Doing so gives us two equations. 9+( 3y-7)i&= 7+x+( -3)i &⇕ 9= 7+x &and 3y-7= -3 Now we can solve these equations one at a time.

The given equation is true when x=2 and y= 43. We can check our answers by substituting these values into the equation and simplifying.

The complex number that results on each side of the equation is 9-3i. Therefore, our solution is correct.

Consider the following complex number

z .

z = - 4 + 7 i

Pair the given complex numbers with the correct definitions.

We are given the following complex number. z=-4+7i Let's find its additive inverse and conjugate.

Additive Inverse of z

The additive inverse of a complex number z is - z.

We have found that the additive inverse of z is 4-7i.

Sum of z and Its Additive Inverse

By the definition of the additive inverse, the sum of a complex number and its additive inverse is 0. Let's show this by calculating z+(- z).

We have found that the sum of z and its additive inverse is 0.

Conjugate of z

The conjugate of a complex number z=a+bi, denoted by z, is written by changing the sign of the imaginary part of z. With this information, we can find the conjugate of z=- 4+7i. z=- 4+ 7i The imaginary part of z is 7. Therefore, the conjugate of z will have - 7 as its imaginary part. z=- 4-7i

Sum of z and Its Conjugate

Let's now calculate the sum of z=-4+7i and its conjugate z=-4-7i.

The sum of z and its conjugate is - 8.

Conclusion

Let's take a final look at all the pairings! 4-7i &→ Additive inverse ofz 0 &→ Sum ofzand its additive inverse -4-7i &→ Conjugate ofz -8 &→ Sum ofzand its conjugate

	18 Theory slides
	14 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

The Basics of Complex Numbers

Catch-Up and Review

Extra

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Extra

Option A

Option B

Option C

Option D

Conclusion

Additive Inverse of z

Sum of z and Its Additive Inverse

Conjugate of z

Sum of z and Its Conjugate

Conclusion

The Basics of Complex Numbers

Recommended exercises