Pure 1.1 - Complex numbers

Question 1

What are imaginary numbers?

Answer 1

A number that can be written as a real number multiplied by the imaginary constant i, they are written in the form 𝑏i, where 𝑏ϵℝ, and the set of imaginary numbers is represented with the symbol 𝕀.

Question 2

What is the imaginary constant?

Answer 2

i=√(-1)

Question 3

What are complex numbers?

Answer 3

Sums of imaginary and real numbers are called complex numbers, they are written in the form 𝑎+𝑏i, where 𝑎, 𝑏ϵℝ, and the set of complex numbers is represented with the symbol ℂ. In complex numbers, Re(𝑧) represents the real part of the number and Im(𝑧) represents the imaginary parts.
They can be added and subtracted from one another by adding or subtracting their real parts and adding or subtracting their imaginary parts.
They can be multiplied by multiplying out their brackets as normal.

Question 4

Describe how complex numbers can be used to find the roots to any quadratic equation with real coefficients.

Answer 4

If the discriminant b²-4ac<0 then the quadratic equation ax²+bx+c=0 has two distinct complex roots, neither of which are real. The roots are the complex conjugate of the other.

Question 5

What is the complex conjugate?

Answer 5

For any complex number 𝑧=𝑎+𝑏i, the complex conjugate of the number is defined as 𝑧*=𝑎-𝑏i.

Question 6

What is the result of 𝑧+𝑧*, where 𝑧=𝑎+𝑏i?

Answer 6

𝑧+𝑧*=𝑎+𝑏i+𝑎-𝑏i= 2𝑎

Question 7

What is realising the denominator?

Answer 7

The process of removing a complex/imaginary number from the denominator of a fraction by multiplying it by its conjugate.

Question 8

Describe the roots of a cubic polynomial, including the possibility of complex roots.

Answer 8

If f(𝑧) is a polynomial with real coefficients and 𝑧₁ is a root of f(𝑧), then f(𝑧₁)=0, so f(𝑧₁*)=0 too.
An equation of the form 𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑=0, a cubic equation, has three roots. For a cubic equation with real coefficients, either:
all three roots are real, or
one root is real and the other two roots form a complex conjugate pair.

Question 9

Describe the roots of a quartic polynomial, including the possibility of complex roots.

Answer 9

If f(𝑧) is a polynomial with real coefficients and 𝑧₁ is a root of f(𝑧), then f(𝑧₁)=0, so f(𝑧₁*)=0 too.
An equation of the form 𝑎𝑥⁴+𝑏𝑥³+𝑐𝑥²+𝑑𝑥+𝑒=0, a quartic equation, has four roots. For a quartic equation with real coefficients, either:
all four roots are real, or
two roots are real and the other two form a complex conjugate pair, or
the four roots form two pairs of complex conjugates pairs.

Question 10

What is the result of 𝑧𝑧*, where 𝑧=𝑎+𝑏i?

Answer 10

𝑧𝑧*=(𝑎+𝑏i)(𝑎-𝑏i)=𝑎²+𝑎𝑏i-𝑎𝑏i-(𝑏i)²=𝑎²+𝑏²