Introduction to Complex Numbers: Lecture 1 - Oxford Mathematics 1st Year Student Lecture

To make sure our students, who come from all over the world, are up to speed for the challenges ahead, this lecture recaps much of the work they should have done at high school, providing a short description of some of the basics of complex numbers.

But even in an recap there are subtle differences and pointers to what's to come.

All first and second year lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.

[toddbussey]

Haven't taken a university math course in 30 years, but I was happy to see that I understood what he was explaining.

[iñigote]

Introduction to Complex Numbers
● [0:14]. Introduction to complex numbers.
● [0:33]. Notation for real numbers (ℝ) and complex numbers (ℂ).
● [1:12]. Definition of a complex number: z ∈ ℂ means that z = a + ib, where a and b are real and i² = -1.
● [1:42]. Real part of a complex number: Re(z) = a.
● [2:01]. Imaginary part of a complex number: Im(z) = b.


Basic Operations with Complex Numbers
● [2:18]. Addition of complex numbers: (a + ib) + (c + id) = (a + c) + i(b + d).
■ The addition of complex numbers is commutative.
● [3:14]. Multiplication of complex numbers: (a + ib)(c + id) = (ac - bd) + i(ad + bc).
■ The multiplication of complex numbers is also commutative.
● [4:31]. Division of complex numbers, z = (a + ib) and w = (c + id) :
■ The conjugate of the denominator is used: (c + id)(c - id) = c² + d².
■ (a + ib) / (c + id) = [(a + ib)(c - id)] / (c² + d²).
■ The expression is simplified to obtain the real and imaginary parts of the result.
● [6:40]. Division is defined as long as the denominator is not zero (w ≠ 0).
● [7:18]. Equality of complex numbers: Two complex numbers are equal if and only if their real and imaginary parts are equal.
● [7:43]. Conjugate of a complex number: z = a + ib, then z̅ = a - ib.
■ The product of a complex number and its conjugate is a non-negative real number: zz̅ = a² + b².
● [9:29]. Real part of z: Re(z) = (z + z̅) / 2.
● [9:39]. Imaginary part of z: Im(z) = (z - z̅) / 2i.
● [10:12]. Properties of the conjugate:
■ (z + w)̅ = z̅ + w̅.
■ (zw)̅ = z̅w̅.
■ (z / w)̅ = z̅ / w̅ (if w ≠ 0).


Complex Numbers and Polynomials
● [11:49]. Motivation for complex numbers: They arise from the need to find roots of polynomials that do not have real solutions.
■ Example: The polynomial x³ + 2x² + 2x + 1 has a real root at x = -1.
■ The other two roots are complex: -1/2 ± i√3 / 2.
● [14:45]. Fundamental Theorem of Algebra: Every polynomial of degree n has n complex roots (counting multiplicities).
● [18:17]. Theorem (unnamed): If all coefficients of a polynomial are real, then:
■ The polynomial can be factored into linear terms corresponding to the real roots (z - α₁, z - α₂, ..., z - αᵣ).
■ And into quadratic factors corresponding to pairs of complex conjugate roots: (z - γ₁)(z - γ̅₁), (z - γ₂)(z - γ̅₂), ..., (z - γₛ)(z - γ̅ₛ).
● [22:15]. Consequence: If a polynomial has real coefficients and γ is a complex root, then its conjugate γ̅ is also a root.
Polar Form of Complex Numbers
● [23:26]. Graphical representation of complex numbers:
■ The horizontal axis represents the real part (x).
■ The vertical axis represents the imaginary part (y).
● [24:46]. Polar form: A complex number z = x + iy can also be represented in polar coordinates (r, θ).
■ r: modulus of z, denoted as |z|, is the distance from the origin to z: |z| = √(x² + y²).
■ θ: argument of z, denoted as arg(z), is the angle between the positive real axis and the line connecting the origin to z.
● [26:46]. The argument is undefined for z = 0.
● [27:22]. Special cases of the argument:
■ If y > 0 and x = 0, then arg(z) = π/2.
■ If y < 0 and x = 0, then arg(z) = 3π/2.
● [27:58]. The argument is periodic: θ, θ + 2π, θ - 2π, θ + 4π, etc., define the same argument.
● [28:59]. The principal value of the argument is in the interval 0 ≤ θ < 2π.
● [29:57]. Properties of the conjugate in polar form:
■ |z̅| = |z|.
■ arg(z̅) = -arg(z).
● [30:47]. Relationships between Cartesian and polar coordinates:
■ x = r cos(θ).
■ y = r sin(θ).
● [31:27]. Proposition:
■ |zw| = |z||w|.
■ |z/w| = |z| / |w| (if w ≠ 0).
■ |z|² = zz̅.
■ arg(zw) = arg(z) + arg(w) (if z ≠ 0 and w ≠ 0).
■ arg(z/w) = arg(z) - arg(w) (if z ≠ 0 and w ≠ 0).
● [35:35]. No information is provided about the modulus or the argument of z + w in this lesson.
● [35:49]. Triangle inequality: |z + w| ≤ |z| + |w|.


Proofs of Some Properties
● [36:41]. Some proofs are omitted, but can be found in the course's online notes.
● [36:49]. Example of proof: (zw)̅ = z̅w̅.
● [39:58]. Example of proof: if a polynomial has real coefficients and γ is a root, then γ̅ is also a root.

[janspl]

Now I can tell my friends that I have attended a lecture at Oxford University.

[mwagon]

you know what? i was amazed to see the whiteboard that could be lifted up😮

[Sunita.Kumari]

I Blood BOILs to see him explain arguement and triangle inequality without explaining the geometrical significance of complex nos.! GO and learn that first!

[HkslrHaks]

Awesome, are you guys going to upload whole series of lectures?

[MuhammadQasim-kjk]

Upload the whole course please ... 💛🙋😊

[TesfaAlemayehu-oc]

I am ethiopia student so the concept of complex nember i was learnet in grade 11 unit seven

[scottychen]

Is that necessarily a plane ?


Obviously, it is an assertorical object that you are symbolizing with ‘ i ‘ .


There are ‘ imaginary sets ‘ that that logically generate topologies that don’t necessarily respect the metric topology of the plane .

[DoutorGutembergYouTuber]

Já estou assistindo desde o primeiro ano do ensino médio

[engineeringmadeasy]

So they start teaching you from Further Maths right?

[dean]

Was this a “free lecture” by any chance? It looked like Tom and Harry felt glad to have attended it..

[AdamJackson-kfeg]

Thank you for this. I love these. Sorry if I'm missing something but I wasn't sure what the notation at 38:43 on the RHS represented?

[freealliance]

Thank You for uploading this i need this for my preparation.

[pelasgeuspelasgeus]

Can someone explain why in the complex plane the imaginary axis unit i=sqrt(-1) is depicted equal to the real axis unit 1?

[Georgexb]

Remember when the great Vicky Neale used to teach this course, RIP

[dschai]

1×1-0×0, 2×2-1×1, 3×3-2×2, 4×4-3×3, 5×5-4×4, 6×6-5×5, 7×7-6×6, 8×8-7×7, 9×9-8×8...odd numbers

[AndreiSfiraiala]

In my time that was taught in high school.

[David-mfg]

Polite correction "principle" should be "principal"

[RchandraMS]

What's the basis for the assumption Z = a + ib?