Complex numbers: operations and the Gauss plane · Grade 12 · 5 units
In Complex numbers: operations and the Gauss plane (Grade 12 · 5 units) you practise 6 skills: The powers of i · Adding, subtracting and multiplying complex numbers · Dividing complex numbers · Equal complex numbers · The modulus and the conjugate · Quadratic equations with complex solutions. Complex numbers were born from the wish to solve equations with no real solution, such as an equation in which the square of a number is minus one. A new number is defined, the imaginary unit i, whose square is minus one. Every complex number is the sum of a real part and an imaginary part, and it is represented by a point in the Gauss plane: the real part on the horizontal axis and the imaginary part on the vertical axis.
Add and subtract complex numbers by their real parts and their imaginary parts separately, and multiply them like algebraic expressions, putting minus one wherever the square of i appears. To divide, multiply the numerator and the denominator by the conjugate of the denominator, so the denominator becomes a real number. Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal. The conjugate of a complex number is the same number with the opposite imaginary part, and its point is the reflection in the real axis. The modulus of a complex number is the distance of its point from the origin, and the modulus of the difference of two numbers is the distance between their points. There is no order on the complex numbers: you cannot say that one complex number is greater than another. A quadratic equation whose expression under the root in the quadratic formula is negative has no real solution, but it has two complex solutions. When the coefficients of the equation are real, these two solutions are conjugates of each other.
Tip: When you divide by a complex number, the conjugate of the denominator turns the denominator into a real number.
What you learn
The powers of i
For example: What is the value of i³ ?
- 1
- i
- −1
- −i
Answer and solution
The answer: −i
The remainder when 3 is divided by four is 3 , so i³ = −i .
Adding, subtracting and multiplying complex numbers
For example: The complex number z = (9 − 7i) − (−1 + 7i) is given. What are the coordinates of the point that represents z in the Gauss plane?
Answer and solution
The answer: (10, −14)
Subtract the real parts and the imaginary parts separately to get z = 10 − 14i . The point that represents it is (10, −14) .
Dividing complex numbers
For example: The complex number z = 1 ÷ (−1 + i) is given. What are the coordinates of the point that represents z in the Gauss plane?
Answer and solution
The answer: (−0.5, −0.5)
The conjugate of the denominator is −1 − i . The denominator becomes 2 and the numerator −1 − i . So z = −0.5 − 0.5i , and the point that represents it is (−0.5, −0.5) .
Equal complex numbers
For example: The real numbers x and y satisfy x + yi = (−3 + 4i)(−3 − i) . What is the point (x, y) ?
Answer and solution
The answer: (13, −9)
Expand the brackets and substitute i² = −1 : the product is 13 − 9i . So x = 13 and y = −9 , and the point is (13, −9) .
The modulus and the conjugate
For example: The complex number z = 5 − 4i is given. What is the modulus of z ?
Answer and solution
The answer: √41
Here |z|² = 25 + 16 = 41 , so |z| = √41 .
Quadratic equations with complex solutions
For example: The equation (x − 4)² = −36 is given. What are the coordinates of the point that represents the solution with a positive imaginary part?
Answer and solution
The answer: (4, 6)
Here x − 4 = 6i or x − 4 = −6i , so x = 4 + 6i or x = 4 − 6i . The solution 4 + 6i has a positive imaginary part, and its point is (4, 6) .
Not on the site yet
Parts of this topic that are not built yet: An order on the complex numbers (there is none) · The fundamental theorem of algebra (enrichment).
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).