The non-negative distance from the origin
Обяснение
The modulus is the distance of the complex number from the origin, so it is always non-negative. The angle with the positive real axis is the argument, not the modulus.
-2 - 3i
Обяснение
The complex conjugate is formed by changing the sign of the imaginary part only. So -2 + 3i becomes -2 - 3i.
0
Обяснение
The four fourth roots of unity cancel in pairs, so their sum is 0. Their product, by contrast, is 1.
-1
Обяснение
For a primitive fourth root of unity, squaring gives -1 and the fourth power returns 1. This matches the fact that a fourth root satisfies z^4=1.
The modulus
Обяснение
In polar form, r is the modulus and θ is the argument. The argument is the angle, not the distance from the origin.
tan^{-1}(y/x)
Обяснение
The angle is given by θ = tan^{-1}(y/x) in the source material. A common mistake is to reverse the ratio.
Each input has exactly one output
Обяснение
A function assigns each input in its domain exactly one output. This is why its graph must pass the vertical line test.
Period
Обяснение
The smallest positive shift that leaves the function unchanged is the period. Any multiple of that period is also a period.
The constant difference between consecutive terms
Обяснение
An arithmetic progression is defined by a constant difference between consecutive terms. A constant ratio would describe a geometric progression.
n/2{2a+(n-1)d}
Обяснение
The standard sum formula for an arithmetic progression is n/2{2a + (n-1)d}. It adds the first and last terms in a structured way across all n terms.
Ordered selections of r objects from n objects
Обяснение
Permutations count ordered selections, so nP_r is about arrangement. Combinations, by contrast, ignore order.
1
Обяснение
By convention, 0! = 1. This makes factorial formulas consistent in combinatorics and algebra.
Factor theorem
Обяснение
The factor theorem directly links a zero of the polynomial to the factor (x - a). The remainder theorem instead says the remainder upon division by (x - a) is f(a).
10
Обяснение
Using the factor theorem with x = 2 gives f(2) = 0, which leads to k = 10. This is a standard application of checking a factor by substitution.
cos(α + β) + cos(α - β)
Обяснение
The source gives cos(α + β) + cos(α - β) = 2cos α cos β. The difference of the cosine terms gives a sine-product identity instead.
180°
Обяснение
Allied angles are angle pairs that add up to 180°. The pair θ and 180° - θ is the standard example.
lim_{x→a} [f(x) - f(a)]/(x - a)
Обяснение
The derivative at a point is the limit of the difference quotient as x approaches a. This is the standard limit definition of the derivative.
1/12
Обяснение
The source states that for f(x) = x^{1/3}, f'(8) = 1/12. This is a direct derivative evaluation at the given point.
When they are perpendicular
Обяснение
An angle of π/2 means the vectors are perpendicular. Parallel vectors have angle 0 or π instead.
The vectors are parallel
Обяснение
If the cross product is zero, the vectors are parallel, provided they are non-zero. The cross product measures the sine of the angle between them.
Запомнете отговорите с 20 флашкарти по Advanced Complex Numbers and Roots.
Complex number — modulus?
Distance from origin in plane.
Complex number — argument?
Angle with positive real axis.
Complex conjugate — change?
Sign of imaginary part.
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