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A Semester of First Year Physics with Peter Eyland
Lecture 12a (Complex Numbers)
In this lecture the following are introduced:
• Real and Imaginary numbers
• Cartesian form for complex numbers
• Basic operations with complex numbers
• Complex conjugates
• Absolute value or modulus
• Polar form for complex numbers
• Exponential form for complex numbers
Real numbers
The real numbers include integers (+1, -2), fractions (+½, -¾) and irrationals (like √2). They can be ordered along a real number axis as shown. |
When a number on the real number axis is multiplied by -1 it ends up rotated anticlockwise by 180^{0}, e.g. |
Imaginary numbers
When a number is squared, the number is multiplied by itself.
Now what number squared will produce -1?
Call the number that does this "j".
Now j^{2} = j×j = -1 = anticlockwise rotation of 180^{0}.
As an example, to change +99 to -99 needs +99×j×j, i.e. 99 times two multiples of "j" gives 180^{0}.
This must mean that one multiple of j is an anticlockwise rotation of 90^{0}.
The imaginary numbers are defined as numbers multiplied by j,
and they lie on an axis rotated at right angles to the real numbers. |
Cartesian form for complex numbers
Complex numbers lie in the plane formed by the real and imaginary axes and are written in Cartesian form as, z = x = j·y. |
Basic Operations with complex numbers
Take two complex numbers: |
Addition: |
Multiplication: |
Division: |
The commutative, associative and distributive laws hold for addition and multiplication.
Complex Conjugates
The complex conjugate is a reflection about the real axis. |
Absolute value or Modulus
The absolute value (or modulus) of a complex number is the straight-line length from the origin to the number's co-ordinate point in the complex plane.
Polar Form for complex numbers
In polar form, z = (r,θ) = r·cosθ + j·r·sinθ. |
Multiplying two complex numbers in Polar form:
The moduli are multiplied and the arguments are added.
Multiplying by j:
Multplying by j causes the expected 90^{0} anticlockwise rotation.
Dividing one complex number by another in Polar form:
One modulus is divided by the other and the arguments are subtracted.
Forming the reciprocal of complex number in Polar form (using the result above):
The modulus is reciprocal and the complex conjugate is formed with the argument.
Exponential Form for complex numbers
The Polar form may now be written as: |
Multiplying and dividing:
This enables the nth roots of a complex number to be calculated and is a form of De Moivre's theorem.
Note that multiplying a complex number by a·e^{jφ} multiplies its modulus by a and rotates it by φ.
Dividing a complex number by a·e^{jφ} shrinks its modulus by 1/a and rotates it by -φ.
In the following π = the Greek letter pi and from the cosine and sine values:
e^{j·(π/2)} = j
e^{j·π} = -j
e^{j·(2π)} = 1, hence for n an integer
e^{(z + j·2πn)} = e^{z}
For the complex conjugate
Summarising:
Imaginary numbers are rotated 90^{0} from the real numbers
Complex numbers are written as:
The commutative, associative and distributive laws hold for addition and multiplication.
Complex Conjugate:
Modulus:
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