Complex numbers Flashcards
What does i represent?
The square root of -1
What is a complex number?
A number in the form a+bi
What is the complex conjugate of a complex number a+bi?
a-bi
How is the complex conjugate of a complex number z notated?
z*
If a complex number z is the root of a polynomial, what is also a root?
z*
What does (a+bi)(a-bi) expand to?
a²+b²
What is an Argand diagram?
A representation of a complex number on a Cartesian coordinate system; the horizontal axis is the real axis, while the vertical axis is the imaginary axis
What is the modulus of a complex number z?
The length of the straight line going from the origin to z on an Argand diagram
How is the modulus of a complex number z notated?
|z|
How can the modulus of a complex number a+bi be calculated?
√(a²+b²)
What is the argument of a complex number z?
The angle between the real axis and z on an Argand diagram
How is the argument of a complex number z notated?
arg(z)
How can the argument of a complex number x+iy be calculated (in radians) when x>0?
arctan(y/x)
How the argument of a complex number x+iy be calculated (in radians) when x<0 and y≥0?
arctan(y/x)+π
How the argument of a complex number x+iy be calculated (in radians) when x<0 and y<0?
arctan(y/x)-π
How the argument of a complex number x+iy be calculated (in radians) when x=0 and y>0?
π/2
How the argument of a complex number x+iy be calculated (in radians) when x=0 and y<0?
-π/2
What is modulus—argument form?
A complex number z can be written in the form r(cos(θ)+i*sin(θ)) where r=|z| and θ=arg(z)
What is |z₁*z₂| equal to?
|z₁|*|z₂|
How can the complex number z satisfying, for example, z²+z=x+iy, where x and y are constants, be found?
Let z=a+bi where a and b are real, such that (a+bi)²+(a+bi)=x+iy. Expand to give a²-b²+2abi+a+bi=x+iy. Then equate real and imaginary parts to give a²-b²+a=x and 2ab+b=y, and then solve the simultaneous equations to find a and b
