Almost everything in mathematics was born out of necessity. When we only knew natural numbers 1, 2, 3, 4…, and whole numbers 0,1,2,3,4,…, and we stumbled upon x + 10 = 0 leading us to the necessity of negative numbers. Similarly, then we stumbled upon 4x + 5 = 0 leading us to the necessity of fractions. Then, we stumbled upon x2 – 8 = 0 leading us to the necessity of irrational numbers. Then, finally we stumbled upon x2 + 1 = 0 leading us to imaginary numbers.
Complex numbers are of the form a + bi where a and b are real numbers and i is an imaginary number such that i2 = -1
The term imaginary numbers is a misnomer. Because they’re kind of named as an opposite of real numbers, but it doesn’t imply they’re unreal. Imaginary numbers are as real as real numbers. Just as negative numbers extend our ability to solve equations and describe concepts like debt, imaginary numbers extend the number system to solve problems where real numbers fall short.
Applications of Complex Numbers:
- Without complex numbers, equations like x2 +1 = 0 have no roots
- Multiplication by an Imaginary Number leads to a rotation in the complex plane. This is a useful way to mathematically represent complex rotations.
- Euler’s formula,
, converts difficult trigonometric integrals and derivatives into simpler exponential ones.
- Complex numbers are crucial for solving linear differential equations with constant coefficients, particularly when dealing with oscillation and resonance, where solutions involve complex roots.
- Used in electrical engineering a lot. Used to determine phases and stability of systems. ejx = cos(x) + j*sin(x). (j’s used, not i’s, since we use i for current). That’s super powerful in electrical engineering.
Calculating the current in a circuit normally involves a messy “differential equation” that uses calculus to account for the resistor, inductor, and capacitor all at once.
Instead of doing that hard math, you can use complex numbers (assuming the circuit is in a steady state). This turns the calculus into simple algebra:
Current × Total Resistance (Impedance) = Voltage
1. The Calculus Approach
This equation describes the voltage across each component (Resistor, Inductor, and Capacitor) over time. It is difficult to solve because it mixes derivatives and integrals:
2. The Complex Algebra (Phasor) Approach
By using complex numbers, we treat everything like a simple resistor. The “complex resistance” is called Impedance (\(Z\)). This turns the calculus into a version of Ohm’s Law (V = I * Z)
Finally, current I is:
Hope this is useful, thank you.
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