Examples of imaginary in the following topics:
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- 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, $i=\sqrt{-1}$.
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- 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
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- 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.
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- 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)
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- 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.
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- 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: $(2+0i)(4-3i)=2(4-3i)=8-6i.$
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- 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.
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- 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.
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- 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 $(a^4-6a^2b^2+b^4)+(4a^3b-4ab^3)i$.
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- Draw an imaginary circular line representing the circular hole in our quiz.
- How does this imaginary circle change as the metal is heated?