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, 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, here's a quick recap of what we covered:
Exercises checked off