imaginary number

Examples of imaginary number 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, $i=\sqrt{-1}$.
  • We can write the square root of any negative number in terms of $i$.

• 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.
  • 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 and Subtraction of Complex Numbers

  • Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately.
  • Complex numbers can be added and subtracted to produce other complex numbers.
  • For example, the sum of $2+3i$ and $5+6i$ can be calculated by adding the two real parts $(2+5)$ and the two imaginary parts $(3+6)$ to produce the complex number $7+9i$.
  • Note that this is always possible since the real and imaginary parts are real numbers, and real number addition is defined and understood.
  • Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately

• Sets of Numbers

  • The set of natural numbers, also known as "counting numbers," includes all whole numbers starting at 1 and then increasing.
  • The set of real numbers includes every number, negative and decimal included, that exists on the number line.
  • The set of rational numbers, denoted by the symbol $\mathbb{Q}$, includes any number that is written as a fraction.
  • The set of imaginary numbers, denoted by the symbol $\mathbb{I}$, includes all numbers that result in a negative number when squared.
  • The set of complex numbers, denoted by the symbol $\mathbb{C}$, includes a combination of real and imaginary numbers in the form of $a+bi$ where $a$ and $b$ are real numbers and $i$ is an imaginary number.

• The Discriminant

  • The number of roots of the function can be determined by the value of $\Delta$.
  • If ${\Delta}$ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.
  • Because adding and subtracting a positive number will result in different values, a positive discriminant results in two distinct solutions, and two distinct roots of the quadratic function.
  • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
  • Explain how and why the discriminant can be used to find the number of real roots of a quadratic equation

• 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 multiplication of two complex numbers is defined by the following formula:
  • 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

  • Two complex numbers can be multiplied to become another complex number.
  • Note that this last multiplication yields a real number, since:
  • 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: $(2+0i)(4-3i)=2(4-3i)=8-6i.$
  • Note that it is possible for two nonreal complex numbers to multiply together to be a real number.

• Complex Conjugates and Division

  • The complex conjugate of the complex number z = x + yi is defined as x - yi.
  • The real and imaginary parts of a complex number can be extracted using the conjugate, respectively:
  • Moreover, a complex number is real if and only if it equals its conjugate.
  • The reciprocal of a nonzero complex number $z = x + yi$ is given by
  • 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

  • Powers of complex numbers can be computed with the the help of the binomial theorem.
  • In what follows, it is useful to keep in mind the powers of the imaginary unit $i$:
  • Powers of complex numbers can be computed with the the help of the binomial theorem.
  • If we gather the real terms and the imaginary terms, we have the complex number:
  • Connect complex numbers raised to a power to the binomial theorem