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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
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Subjects
:
clep
,
math
Instructions:
Answer 50 questions in 15 minutes.
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.
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 Numbers: Add & subtract
four different numbers: i - -i - 1 - and -1.
radicals
Polar Coordinates - Multiplication by i
2. y / r
radicals
Polar Coordinates - sin?
the vector (a -b)
Complex Subtraction
3. 1
Subfield
Complex Exponentiation
i^4
How to find any Power
4. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
the complex numbers
We say that c+di and c-di are complex conjugates.
Real Numbers
conjugate
5. R^2 = x
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number
Square Root
i^2
6. z1z2* / |z2|²
How to solve (2i+3)/(9-i)
non-integers
Rational Number
z1 / z2
7. The square root of -1.
Imaginary Unit
irrational
Subfield
cosh²y - sinh²y
8. The complex number z representing a+bi.
Imaginary number
How to solve (2i+3)/(9-i)
Affix
|z| = mod(z)
9. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
a + bi for some real a and b.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cos iy
Integers
10. Divide moduli and subtract arguments
z - z*
Polar Coordinates - Division
Square Root
natural
11. When two complex numbers are multipiled together.
Complex Multiplication
point of inflection
We say that c+di and c-di are complex conjugates.
Affix
12. 1st. Rule of Complex Arithmetic
i^2 = -1
non-integers
adding complex numbers
cos z
13. Written as fractions - terminating + repeating decimals
Polar Coordinates - r
irrational
rational
Subfield
14. A subset within a field.
Subfield
the distance from z to the origin in the complex plane
x-axis in the complex plane
Real and Imaginary Parts
15. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
standard form of complex numbers
rational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
16. Numbers on a numberline
point of inflection
integers
(cos? +isin?)n
Rational Number
17. V(zz*) = v(a² + b²)
irrational
'i'
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z| = mod(z)
18. ½(e^(-y) +e^(y)) = cosh y
transcendental
standard form of complex numbers
cos iy
i^2
19. 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.'
zz*
Irrational Number
Complex Number
complex numbers
20. 1
cosh²y - sinh²y
0 if and only if a = b = 0
rational
adding complex numbers
21. A complex number may be taken to the power of another complex number.
Argand diagram
i^2
Complex Numbers: Add & subtract
Complex Exponentiation
22. 2ib
the distance from z to the origin in the complex plane
transcendental
cos iy
z - z*
23. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - Division
Roots of Unity
Integers
Irrational Number
24. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
|z-w|
How to multiply complex nubers(2+i)(2i-3)
Irrational Number
25. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
the complex numbers
complex numbers
four different numbers: i - -i - 1 - and -1.
Imaginary number
26. A number that can be expressed as a fraction p/q where q is not equal to 0.
conjugate pairs
i^4
ln z
Rational Number
27. To simplify a complex fraction
Field
Complex Number Formula
-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
28. 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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29. 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
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.
adding complex numbers
v(-1)
30. 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
has a solution.
cos iy
Rules of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
31. Derives z = a+bi
Liouville's Theorem -
Imaginary Unit
e^(ln z)
Euler Formula
32. Any number not rational
a + bi for some real a and b.
irrational
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
33. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
cos z
i²
Polar Coordinates - Multiplication
34. 2a
complex
z + z*
sin z
i²
35. 3rd. Rule of Complex Arithmetic
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0
Square Root
adding complex numbers
36. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
How to multiply complex nubers(2+i)(2i-3)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Subtraction
sin z
37. R?¹(cos? - isin?)
x-axis in the complex plane
conjugate pairs
Absolute Value of a Complex Number
Polar Coordinates - z?¹
38. All numbers
Polar Coordinates - z?¹
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
cosh²y - sinh²y
39. 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
the distance from z to the origin in the complex plane
Complex Numbers: Add & subtract
cosh²y - sinh²y
Absolute Value of a Complex Number
40. Multiply moduli and add arguments
the vector (a -b)
z1 ^ (z2)
Polar Coordinates - Multiplication
(a + c) + ( b + d)i
41. Cos n? + i sin n? (for all n integers)
Rules of Complex Arithmetic
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.
(cos? +isin?)n
42. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
cos z
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers
43. 1
De Moivre's Theorem
i^2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin iy
44. We can also think of the point z= a+ ib as
(cos? +isin?)n
non-integers
Subfield
the vector (a -b)
45. x / r
e^(ln z)
Polar Coordinates - cos?
interchangeable
i^2
46. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
z - z*
Field
Polar Coordinates - Division
z1 ^ (z2)
47. No i
x-axis in the complex plane
transcendental
real
Every complex number has the 'Standard Form': a + bi for some real a and b.
48. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
subtracting complex numbers
Square Root
(cos? +isin?)n
49. ? = -tan?
Polar Coordinates - Arg(z*)
Euler Formula
four different numbers: i - -i - 1 - and -1.
'i'
50. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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