complex numbers

(noun)

Numbers that have an imaginary part. Usually represented as i.

Related Terms

Examples of complex numbers in the following topics:

  • Complex numbers and constant coefficient differential equations

    • Here we see the main reason for complex numbers.
    • Complex numbers are things of the form $a + i b$ where $a$ and $b$ are real numbers.
    • Addition of complex numbers is component-wise: $(a + i b) + (p + i q) = (a + p) + i(b + q)$.
    • And just as there is an equivalence between Cartesian and polar coordinates, so we can give a "polar" representation of every complex number.
    • Figure 1.5: Every complex number can be represented as a point in the complex plain.

  • Phasors

    • Complex numbers play an important role in physics.
    • Usually, complex numbers are written in terms of their real part plus the imaginary part.
    • For example, $a + bi$ where a and b are real numbers, and $i$ signals the imaginary part.
    • However, it is often practical to write complex numbers in the form of an exponential called a phasor.
    • a complex, time-varying signal may be represented as the product of a complex number that is independent of time and a complex signal that is dependent on time.

  • The Wave Function

    • Typically, its values are complex numbers.
    • Although ψ is a complex number, |ψ|2 is a real number and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured.
    • Furthermore, when we use the wave function to calculate an observation of the quantum system without meeting these requirements, there will not be finite or definite values to use (in this case the observation can take a number of values and can be infinite).

  • Forced motion with damping

    • We can proceed just as before with the undamped, forced oscillations but the algebra is greatly simplified if we use complex numbers.
    • In order for two complex numbers to be equal, their real and imaginary parts must be equal separately.
    • Therefore the real part of the complex "displacement" must satisfy Equation 1.1.23, which is what was claimed.
    • So let's write the complex force as $F = \hat{F} e^{i \omega t}$ and the complex displacement as $x = \hat{x} e^{i \omega t}$ .

  • Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams

    • The key idea in the phasor representation is that a complex, time-varying signal may be represented as the product of a complex number (that is independent of time) and a complex signal (that is dependent on time).
    • For example, we can represent $A\cdot \cos(2\pi \nu t + \theta)$ simply as a complex constant, $A e^{i\theta}$ .
    • A phasor can be seen as a vector rotating about the origin in a complex plane.

  • The Linear Algebra of the DFT

    • $N$ is fixed, that's just the number of data points.
    • The matrix appearing in Equation 4.6.3 is the complex conjugate of $Q$ ; i.e., $Q^*$ .
    • For complex matrices we need to generalize this definition slightly; for complex matrices we will say that $A$ is orthogonal if $(A^T)^* A = A (A^T)^* = I$ .
    • (Technically such a matrix is called Hermitian or self-adjoint--the operation of taking the complex conjugate transpose being known at the adjoint--but we needn't bother with this distinction here. ) In our case, since $Q$ is obviously symmetric, we have:

  • Electric Charge in the Atom

    • Atoms contain negatively charged electrons and positively charged protons; the number of each determines the atom's net charge.
    • The number of protons in an atom defines the identity of the element (an atom with 1 proton is hydrogen, for example, and an atom with two protons is helium).
    • As such, protons are relatively stable; their number rarely changes, only in the instance of radioactive decay.
    • The electrons cloud patterns are extremely complex and is of no importance to the discussion of electric charge in the atom.
    • In the ground state, an atom will have an equal number of protons and electrons, and thus will have a net charge of 0.

  • Superposition and orthogonal projection

    • In general, the sum will require an infinite number of coefficients $c_i$ , since a function has an infinite amount of information.
    • Of course, you can easily think of functions for which all but a finite number of the coefficients will be zero; for instance, the sum of a finite number of sinusoids.
    • Now we simply need to show that the sines and cosines (or complex exponentials) are orthogonal.
    • (If you get stuck, the proof can be found in Boyce and DiPrima, Elementary differential equations and boundary value problems, Chapter 10. ) A similar result holds for the complex exponential, where we define the basis functions as $\xi _ k (x) = e^{i k \pi x/l}$ .

  • Impedance

    • The phase of the complex impedance is the phase shift by which the current is ahead of the voltage.
    • Letting the voltage be a complex exponential we have $i = j \omega CV e^{j \omega t}$.
    • The amplitude of this complex exponential is $I = j \omega CV$.
    • This quantity is known as the element's (complex) impedance.
    • The (real value) impedance is the real part of the complex impedance Z.

  • Van de Graff Generators

    • The separation of charge in a Van de Graaff generator is a complex, multistep process.
    • Numbers in the diagram indicate: 1) hollow metal sphere; 2) upper electrode; 3) upper roller (for example an acrylic glass); 4) side of the belt with positive charges; 5) opposite side of the belt with negative charges; 6) lower roller (metal); 7) lower electrode (ground); 8) spherical device with negative charges, used to discharge the main sphere; 9) spark produced by the difference of potentials