imaginary unit

(noun)

A complex number, usually denoted with i, that is defined as i^2 = -1

Related Terms

  • parallelogram
  • complex numbers

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 i2=−1i^2=-1i​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(ac - bd) + (bc + ad)i(ac−bd)+(bc+ad)i (by the fundamental property of the imaginary unit)
  • Introduction to Complex Numbers

    • A complex number has the form a+bia+bia+bi, where aaa and bbb are real numbers and iii is the imaginary unit.
    • A complex number is a number that can be put in the form a+bia+bia+bi where aaa and bbb are real numbers and iii is called the imaginary unit, where i2=−1i^2=-1i​2​​=−1.
    • In this expression, aaa is called the real part and bbb the imaginary part of the complex number.
    • To indicate that the imaginary part of 4−5i4-5i4−5i is −5-5−5, we would write Im{4−5i}=−5\text{Im}\{4-5i\} = -5Im{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.
  • 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 000 is easily multiplied as this example shows: (2+0i)(4−3i)=2(4−3i)=8−6i.(2+0i)(4-3i)=2(4-3i)=8-6i.(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 iii:
    • If we gather the real terms and the imaginary terms, we have the complex number:
  • 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, iii, is defined as the square root of -1: thus, i=−1i=\sqrt{-1}i=√​−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.
    • we would compute 4−24-24−2 obtaining 222 for the real part, and calculate −3−4=−7-3-4=-7−3−4=−7 for the imaginary part.
    • Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately
  • 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.
  • The Discriminant

    • If Δ{\Delta}Δ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.
    • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
  • Sets of Numbers

    • The set of imaginary numbers, denoted by the symbol I\mathbb{I}I, includes all numbers that result in a negative number when squared.
    • The set of complex numbers, denoted by the symbol C\mathbb{C}C, includes a combination of real and imaginary numbers in the form of a+bia+bia+bi where aaa and bbb are real numbers and iii is an imaginary number.
  • Defining Trigonometric Functions on the Unit Circle

    • In this section, we will redefine them in terms of the unit circle.
    • Recall that a unit circle is a circle centered at the origin with radius 1.
    • The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram.
    • We can find the coordinates of any point on the unit circle.
    • The unit circle demonstrates the periodicity of trigonometric functions.
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