Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:
For example,
is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
Nice work completing Cartesian form and Argand diagrams, here's a quick recap of what we covered:
Exercises checked off
Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:
For example,
is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
Nice work completing Cartesian form and Argand diagrams, here's a quick recap of what we covered:
Exercises checked off