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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
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Study First
Subjects
:
clep
,
math
Instructions:
Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can
study here
.
Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.
This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.
1. All the powers of i can be written as
Complex Addition
point of inflection
Field
four different numbers: i - -i - 1 - and -1.
2. I = imaginary unit - i² = -1 or i = v-1
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers
Absolute Value of a Complex Number
Polar Coordinates - cos?
3. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Complex Number
subtracting complex numbers
cos z
Square Root
4. 1
i^2
e^(ln z)
0 if and only if a = b = 0
sin iy
5. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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6. 5th. Rule of Complex Arithmetic
standard form of complex numbers
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection
7. Root negative - has letter i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
imaginary
integers
8. All numbers
i²
z1 / z2
complex
sin z
9. 3
Polar Coordinates - Arg(z*)
(cos? +isin?)n
i^3
Absolute Value of a Complex Number
10. V(zz*) = v(a² + b²)
|z| = mod(z)
sin iy
Subfield
Real and Imaginary Parts
11. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.
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12. No i
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.
cosh²y - sinh²y
real
13. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
We say that c+di and c-di are complex conjugates.
irrational
Complex Number
14. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real and Imaginary Parts
z - z*
Affix
15. Any number not rational
Argand diagram
irrational
Polar Coordinates - z
Complex Addition
16. V(x² + y²) = |z|
Complex Number Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
Complex Number
17. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Number
Roots of Unity
radicals
The Complex Numbers
18. Divide moduli and subtract arguments
Polar Coordinates - Division
rational
i^1
the complex numbers
19. Has exactly n roots by the fundamental theorem of algebra
Complex numbers are points in the plane
De Moivre's Theorem
Complex Exponentiation
Any polynomial O(xn) - (n > 0)
20. Real and imaginary numbers
|z| = mod(z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
conjugate pairs
21. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z1 ^ (z2)
conjugate pairs
conjugate
complex
22. R^2 = x
Argand diagram
The Complex Numbers
Complex numbers are points in the plane
Square Root
23. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0
Absolute Value of a Complex Number
subtracting complex numbers
24. Not on the numberline
Complex Number
Complex Multiplication
non-integers
i^1
25. 1
-1
Integers
a + bi for some real a and b.
i^4
26. ? = -tan?
Complex Multiplication
transcendental
Polar Coordinates - Arg(z*)
Real Numbers
27. x / r
Polar Coordinates - cos?
Real and Imaginary Parts
Complex Division
Polar Coordinates - sin?
28. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - Multiplication by i
Integers
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - r
29. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Absolute Value of a Complex Number
Polar Coordinates - sin?
30. 2ib
z - z*
zz*
Irrational Number
Complex Subtraction
31. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
multiplying complex numbers
Polar Coordinates - sin?
Complex Number
subtracting complex numbers
32. Written as fractions - terminating + repeating decimals
Real and Imaginary Parts
v(-1)
Complex Exponentiation
rational
33. In this amazing number field every algebraic equation in z with complex coefficients
real
i²
has a solution.
Polar Coordinates - r
34. A² + b² - real and non negative
zz*
standard form of complex numbers
'i'
four different numbers: i - -i - 1 - and -1.
35. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Rules of Complex Arithmetic
a + bi for some real a and b.
transcendental
adding complex numbers
36. A+bi
Complex Number Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Liouville's Theorem -
Absolute Value of a Complex Number
37. To simplify a complex fraction
Polar Coordinates - sin?
Polar Coordinates - Multiplication
z - z*
multiply the numerator and the denominator by the complex conjugate of the denominator.
38. E ^ (z2 ln z1)
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
cos z
The Complex Numbers
39. A plot of complex numbers as points.
Irrational Number
Argand diagram
De Moivre's Theorem
multiply the numerator and the denominator by the complex conjugate of the denominator.
40. Multiply moduli and add arguments
Polar Coordinates - Multiplication
the complex numbers
conjugate pairs
four different numbers: i - -i - 1 - and -1.
41. E^(ln r) e^(i?) e^(2pin)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)
v(-1)
Polar Coordinates - r
42. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Euler Formula
i²
Polar Coordinates - cos?
Rules of Complex Arithmetic
43. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Number Formula
multiplying complex numbers
Square Root
44. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers
Complex Numbers: Add & subtract
Rules of Complex Arithmetic
45. R?¹(cos? - isin?)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
natural
i^0
46. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
How to find any Power
z1 ^ (z2)
Polar Coordinates - z
Complex Number Formula
47. ½(e^(-y) +e^(y)) = cosh y
a + bi for some real a and b.
'i'
subtracting complex numbers
cos iy
48. I
four different numbers: i - -i - 1 - and -1.
v(-1)
sin z
complex numbers
49. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - r
the distance from z to the origin in the complex plane
Affix
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
50. (e^(iz) - e^(-iz)) / 2i
Absolute Value of a Complex Number
How to multiply complex nubers(2+i)(2i-3)
x-axis in the complex plane
sin z