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12 Cards in this Set
- Front
- Back
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The Unit Imaginary Number i is a number whose square is -i. That is,
i^2 = -i; or i = sqrt (-1). So, what is sqrt(-5)? |
sqrt (-5) = i sqrt (5)
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What is a complex number? What is the real part and what is the imaginary part?
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A complex number is a number of the form a + bi, the real number a is called the real part of a + bi, the real number b is called the imaginary part of a + bi, and i is sqrt (-1).
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Complex numbers can be plotted on a complex number plane with the real part on the abscissa (where x goes) and imaginary part on the ordinate (where y goes). Therefore, the set of real numbers and the set of imaginary numbers are a subset of the set of ???
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Both real and imaginary numbers are subsets of the set of complex numbers since they both lie in the complex plane.
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The number 0 lies on both the real and imaginary number lines, so is it real or imaginary???
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The number zero is in fact both!
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What are complex conjugates?
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The complex numbers a + bi and a - bi are called complex conjugates of each other.
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T/F The product of two complex conjugates is a real number
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True.
(a + bi)(a - bi) = a^2 + b^2 |
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How about when you want to find a quadratic equation from their solutions? Note that the solutions are complex conjugates of each other. You can imagine the form
(x - s1)(x - s2) = 0. Thus what are the key equations to find a, b and c in the quadratic equation? This is when you are given s1 and s2. |
s1s2 = c/a and -(s1 + s2) = b/a
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How do you define the roots of an equation?
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A root of an equation is a solution of that equation.
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When does a quadratic equation have a complex conjugate for a solution?
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If a quadratic equation with real coefficients has a negative discriminants, then the two solutions are complex conjugates of each other.
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What it the remainder theorem and what can it be used for?
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The remainder theorem is if P(x) is a polynomial, then P(b) is equal to the remainder when P(x) is divided by x-b.
It can be used as synthetic substitution technique. For example you can find all values of x that make P(x) = 0. |
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What is the definition of a zero of a function?
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A zero of a polynomial P(x) is a value of x, real or complex, which makes P(x) = 0.
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If P(x) is an nth degree polynomial, then P(x) has exactly n linear factors.And that leads to the fundamental theorem of Algebra which states that if you allow zeros to be complex numbers, then a polynomial P(x) has a least how many zeros?
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A polynomial P(x) has at least one zero, if you allow zeros to be complex numbers.
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