imaginary unit

Examples of imaginary unit in the following topics:

• 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)

• 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.
  • A complex number is a number that can be put in the form $a+bi$ where $a$ and $b$ are real numbers and $i$ is called the imaginary unit, where $i^2=-1$.
  • 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.

• Impedance

  • where V is the amplitude of the AC voltage, j is the imaginary unit (j2=-1), and $\omega$ is the angular frequency of the AC source.
  • The imaginary unit is given the symbol "j", not the usual "i".

• 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: $(2+0i)(4-3i)=2(4-3i)=8-6i.$

• 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 $(a^4-6a^2b^2+b^4)+(4a^3b-4ab^3)i$.

• 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

• 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?
  • Ignoring pressure, we may write: $\alpha_A = \frac{1}{A} \frac{dA}{dT}$, where is some area of interest on the object, and dA/dT is the rate of change of that area per unit change in temperature.

• Phasors

  • Where $_j$ is imaginary.
  • 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.
  • In summary, the parameters that determine a cosinusoidal signal have the following units:

• Hyperbolic Functions

  • Just as the points ($\cos t$, $\sin t$) form a circle with a unit radius, the points ($\cosh t$, $\sinh t$) form the right half of the equilateral hyperbola.
  • In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine.