Complex Numbers

A Guide to Understand

Complex Numbers

Index

1. What are complex numers

2. Addition

3. Subtraction

4. Multiplication

5. Properties of complex numners

6. Division

7. Geometric Interpretation

8. Complex numbers in Quadratics

What are Complex Numbers?

They are the composition of a Real and an Imaginary number

z = x + yi

z is the complex number

x is the real number

yi is the imaginary number

But what is an imaginary number?

A number that is expressed in terms of the square root of a negative number

Since there are no real numbers that are the square roots of negatives, the imaginary number i was created

i² = −1

So, yi means that y will stand for a variable such as 2, 3, 15, etc while including the i. Example = 2i, 3i, 15i

7i² = −7

IMPORTANT: i isn't a variable, it doesnt stand for anything else than the square root of -1, so 5i is the square root of -5.

Addition

• Addition of complex numbers works exactly as monomials or binomials would, as if i was a variable (even if it isn't) but then the other part of the imaginary numbers the "real numbers" gets added normally while the i stays the same.

• Example:

2i + 5i = 7i(7 + 8i) + (11i + 2) = 9 + 19i Imaginary numbers don't get added with real numbers (3 + 2i is not 5i)

Subtraction

• Subtraction of complex numbers works exactly as monomials or binomials would, as if i was a variable (even if it isn't) but then the other part of the imaginary numbers the "real numbers" gets subtracted normally while the i stays the same.

• Example:

5i - 2i = 3i14i - 20i = -6i Imaginary numbers dont get subtracted with real numbers (3 - 2i is not i)

Multiplication

• Multiplication of complex numbers works exactly as monomials or bonimials would, as if i was a variable (even if it isn't) but then the other part of the imaginary numbers the "real numbers" gets multiplied normally while the i stays the same.

• Example:

i * i * i = i³ 2i * 5i = 10i²3 * 3i = 9i (3 + 2i) (-2 + 7i) = - 6 + 21i - 4i + 14i² - 6 + 21i - 4i - 14 - 20 +17

• Note: Since i² = -1, then 14i² will be 14(-1) or - 14

Properties of Complex

• Norm of a Complex ----> Magnitude of the Complex
• To get the magnitude of a complex we have to ignore the i of the imaginary number and apply the pythagorean theorem by adding the square of both parts of the complex number (x² + y² from x + yi). Then we'll get the square of z, which we'll have to use square root to find z. That will be the magnitude, and always will be positive.

Example = 3 + 2i; 3² + 2²; 9 + 4; 13; |3.6|

• Conjugate = Changing by the opposite sign of the imaginary part of the complex
• z = 3 + 2i
• z = 3 - 2i (conjugate of a number)

Division

• For Dividing Complex numbers we have to multiply both numerator and denominator of the fraction by the conjugate of the same demoninator of the fraction.

• Example:

3 + 5i/1 - 2i = (3 + 5i/1 - 2i) (1 + 2i/1 + 2i) = -7 + 11i/5 = -7/5 + (11/5)i

• DENOMINATOR:

• NOMINATOR:

• (1 - 2i) (1 + 2i) = 1 + 2i - 2i - 4i²

= 1 - 4i² = 5

• (3 + 5i) (1 + 2i) = 3 + 6i + 5i + 10i²

= 3 + 11i - 10 = -7 + 11i

• Results of divisions should always be simplified (if possible) or at least separated so that the difference between the real and the imaginary number as part of a new complex number is clear.

Geometric Interpretation

Addition & Subtraction

Multiplication

• The Axis when graphing complex numbers are different, although the negatives still work in the same way (downwards and to the left), instead of y-axis, its the 'i-axis' representing the imaginary numbers inside the complex (the real ones are represented in the x-axis as they normally are).

How to use Complex Numbers to Find all the Roots of a Quadratic Function

We can change the fuction to a quadratic equation by setting equal to zero. Example: 0 = x² + 5x + 6, and later using the quadratic formula to find the two solutions (Positive & Negative)

We know that a quadratic function is is one of the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola.

As stated before, sometimes we find negative square roots inside the formula, and the only way to find a solution is to include the imaginary number 'i'.

Metacognition Document (Plan of Product Unit 2)

https://gimnasiocampestre-my.sharepoint.com/:w:/g/personal/alejandro_pelaez_campestre_edu_co/ETnHfCf3FF9Mi7cpaBmLS30BTmzwzvjB_KypcMIPSEv0Kw?e=syXd4r