Examples of imaginary unit in the following topics:
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- Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule i2=−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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- 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 i2=−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 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.
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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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- 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:
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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=√−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.
- 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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- 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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- If 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.
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- The set of imaginary numbers, denoted by the symbol I, includes all numbers that result in a negative number when squared.
- The set of complex numbers, denoted by the symbol 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.
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- 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.