Complex numbers

Question 1

What does i represent?

Answer 1

The square root of -1

Question 2

What is a complex number?

Answer 2

A number in the form a+bi

Question 3

What is the complex conjugate of a complex number a+bi?

Answer 3

a-bi

Question 4

How is the complex conjugate of a complex number z notated?

Answer 4

z*

Question 5

If a complex number z is the root of a polynomial, what is also a root?

Answer 5

z*

Question 6

What does (a+bi)(a-bi) expand to?

Answer 6

a²+b²

Question 7

What is an Argand diagram?

Answer 7

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

Question 8

What is the modulus of a complex number z?

Answer 8

The length of the straight line going from the origin to z on an Argand diagram

Question 9

How is the modulus of a complex number z notated?

Answer 9

|z|

Question 10

How can the modulus of a complex number a+bi be calculated?

Answer 10

√(a²+b²)

Question 11

What is the argument of a complex number z?

Answer 11

The angle between the real axis and z on an Argand diagram

Question 12

How is the argument of a complex number z notated?

Answer 12

arg(z)

Question 13

How can the argument of a complex number x+iy be calculated (in radians) when x>0?

Answer 13

arctan(y/x)

Question 14

How the argument of a complex number x+iy be calculated (in radians) when x<0 and y≥0?

Answer 14

arctan(y/x)+π

Question 15

How the argument of a complex number x+iy be calculated (in radians) when x<0 and y<0?

Answer 15

arctan(y/x)-π

Question 16

How the argument of a complex number x+iy be calculated (in radians) when x=0 and y>0?

Answer 16

π/2

Question 17

How the argument of a complex number x+iy be calculated (in radians) when x=0 and y<0?

Answer 17

-π/2

Question 18

What is modulus—argument form?

Answer 18

A complex number z can be written in the form r(cos(θ)+i*sin(θ)) where r=|z| and θ=arg(z)

Question 19

What is |z₁*z₂| equal to?

Answer 19

|z₁|*|z₂|

Question 20

How can the complex number z satisfying, for example, z²+z=x+iy, where x and y are constants, be found?

Answer 20

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