LECTURE 18 MARCH 2024- LEVEL 3 -2 Complex Numbers.ppsx
- 1. Complex Numbers Complex : means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined Together) OR A Complex Number is a combination of a Real Number and an imaginary Number .
- 3. Properties The letter z is often used for a complex z = z + bi Z is a Complex Number ; a and b are Real Numbers ; I is the unit imaginary number = . −𝟏. Refer to the real part and imaginary part using Re and Im like 𝒁 𝒎𝒆𝒂𝒏𝒔 𝒄𝒐𝒏𝒋𝒖𝒈𝒂𝒕𝒆 𝒊𝒕 𝒄𝒉𝒂𝒂𝒏𝒈𝒆𝒔 𝒕𝒉𝒆 𝒔𝒊𝒈𝒏 𝒊𝒏 𝒕𝒉𝒆 𝒎𝒊𝒅𝒅𝒍𝒆 𝒐𝒇 𝒁 𝒊𝒔 𝒔𝒉𝒐𝒘𝒏 𝒘𝒊𝒕𝒉 𝒂 𝒔𝒕𝒂𝒓 𝑻𝒉𝒆 𝒄𝒐𝒏𝒋𝒖𝒈𝒂𝒕𝒆 𝒊𝒕 𝒄𝒉𝒂𝒏𝒈𝒆𝒔 𝒕𝒉𝒆 𝒔𝒊𝒈𝒏 𝒊𝒏 𝒕𝒉𝒆 𝒎𝒊𝒅𝒅𝒍𝒆 𝒐𝒇 𝒛 𝒊𝒔 𝒔𝒉𝒐𝒘𝒏 𝒘𝒊𝒕𝒉 𝒂 𝒔𝒕𝒂𝒓: 𝒁 = 𝒂 − 𝒃𝒊
- 4. We can also use angle and distance like this (called polar form): So the complex number 3 + 4i can also be shown as distance 5 and angle 0.927 radians. To convert from one form to the other use Cartesian to Polar conversion. The magnitude of z is: |z| = √(a2 + b2) e can also use angle and distance like this (called polar form): And the angle of z, also called is: Arg(z) = tan-1(b/a) (for a>0) Adding ADDING : To add two complex numbers we add each part separately: (a+bi) + (c+di) = (a+c) + (b+d)i Example: add the complex numbers 3 + 2i and 1 + 7i • add the real numbers, and • add the imaginary numbers: (3 + 2i) + (1 + 7i) = 3 + 1 + (2 + 7)i = 4 + 9i Let's try another: Example: add the complex numbers 3 + 5i and 4 − 3i (3 + 5i) + (4 − 3i) = 3 + 4 + (5 − 3)i = 7 + 2i
- 5. MULLTIPLICATION : To multiply complex numbers: Each part of the first complex number gets multiplied by each part of the second complex number Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" (see Binomial Multiplication for more details): Firsts: a × c Outers: a × di Inners: bi × c Lasts: bi × di (a+bi)(c+di) = ac + adi + bci + bdi2 EXAMPLE Example: (3 + 2i)(1 + 7i) (3 + 2i)(1 + 7i)= 3×1 + 3×7i + 2i×1+ 2i×7i = 3 + 21i + 2i + 14i2 = 3 + 21i + 2i − 14 (because i2 = −1) = −11 + 23i And this: Example: (1 + i)2 (1 + i)(1 + i)= 1×1 + 1×i + 1×i + i2 = 1 + 2i − 1 (because i2 = −1) = 0 + 2i
- 6. Example: i2 We can write i with a real and imaginary part as 0 + i i2= (0 + i)2 = (0 + i)(0 + i) = (0×0 − 1×1) + (0×1 + 1×0)i = −1 + 0i = −1 Conjugates We will need to use conjugates in a minute! - A conjugate is where we change the sign in the middle like this: Example: 5 − 3i = 5 + 3i Multiply both Top and Bottom by conjugate of the bottom DIVIDING
- 7. Graphing of Complex Numbers The complex number consists of a real part and an imaginary part, which can be considered as an ordered pair (Re(z), Im(z)) and can be represented as coordinates points in the euclidean plane. The euclidean plane with reference to complex numbers is called the complex plane or the Argand Plane, named after Jean-Robert Argand. The complex number z = a + ib is represented with the real part - a, with reference to the x- axis, and the imaginary part-ib, with reference to the y-axis. Let us try to understand the two important terms relating to the representation of complex numbers in the argand plane. The modulus and the argument of the complex number. Modulus of the Complex Number The distance of the complex number represented as a point in the argand plane (a, ib) is called the modulus of the complex number. This distance is a linear distance from the origin (0, 0) to the point (a, ib), and is measured as r = |√a2+b2�2+�2|. Further, this can be understood as derived from the Pythagoras theorem, where the modulus