imaginary

Examples of imaginary in the following topics:

• Imaginary Numbers

  • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
  • That is where imaginary numbers come in.
  • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.
  • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.

• Addition and Subtraction of Complex Numbers

  • Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately.
  • This is done by adding the corresponding real parts and the corresponding imaginary parts.
  • The key again is to combine the real parts together and the imaginary parts together, this time by subtracting them.
  • Thus to compute $(4-3i)-(2+4i)$ we would compute $4-2$ obtaining $2$ for the real part, and calculate $-3-4=-7$ for the imaginary part.
  • Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately

• Introduction to Complex Numbers

  • A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
  • In this expression, $a$ is called the real part and $b$ the imaginary part of the complex number.
  • To indicate that the imaginary part of $4-5i$ is $-5$, we would write $\text{Im}\{4-5i\} = -5$.
  • A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.
  • The complex number $-2+3i$ is plotted in the complex plane, 2 to the left on the real axis, and 3 up on the imaginary axis.

• Addition, Subtraction, and Multiplication

  • Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule $i^2=-1$.
  • Complex numbers are added by adding the real and imaginary parts of the summands.
  • The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit.
  • = $(ac - bd) + (bc + ad)i$ (by the fundamental property of the imaginary unit)

• Quantum Tunneling

  • If an object lacks enough energy to pass through a barrier, it is possible for it to "tunnel" through imaginary space to the other side.
  • They never exist in the nodal area (this is forbidden); instead they travel through imaginary space.
  • Imaginary space is not real, but it is explicitly referenced in the time-dependent Schrödinger equation, which has a component of $i$ (the square root of $-1$, an imaginary number):
  • And because all matter has a wave component (see the topic of wave-particle duality), all matter can in theory exist in imaginary space.

• Multiplication of Complex Numbers

  • Note that the FOIL algorithm produces two real terms (from the First and Last multiplications) and two imaginary terms (from the Outer and Inner multiplications).
  • Similarly, a number with an imaginary part of $0$ is easily multiplied as this example shows: $(2+0i)(4-3i)=2(4-3i)=8-6i.$

• Discrete Fourier Transform Examples

  • Figures 4.10 and 4.11 show the real (left) and imaginary (right) parts of six time series that resulted from inverse DFT'ing an array $H_n$ which was zero except at a single point (i.e., it's a Kronecker delta: $H_i = \delta _{i,j} =1$ and zero otherwise; here a different $j$ is chosen for each plot).
  • The real (left) and imaginary (right) parts of three length 64 time series, each associated with a Kronecker delta frequency spectrum.

• Complex Conjugates and Division

  • The real and imaginary parts of a complex number can be extracted using the conjugate, respectively:
  • Neither the real part c nor the imaginary part d of the denominator can be equal to zero for division to be defined.

• Complex Numbers and the Binomial Theorem

  • In what follows, it is useful to keep in mind the powers of the imaginary unit $i$.
  • If we gather the real terms and the imaginary terms, we have the complex number $(a^4-6a^2b^2+b^4)+(4a^3b-4ab^3)i$.

• Area Expansion

  • Draw an imaginary circular line representing the circular hole in our quiz.
  • How does this imaginary circle change as the metal is heated?