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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
Start Test
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. xpressions such as ``the complex number z'' - and ``the point z'' are now
multiplying complex numbers
complex numbers
interchangeable
Polar Coordinates - z?¹
2. The complex number z representing a+bi.
Affix
a + bi for some real a and b.
subtracting complex numbers
z1 / z2
3. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
adding complex numbers
has a solution.
conjugate pairs
4. A number that cannot be expressed as a fraction for any integer.
How to add and subtract complex numbers (2-3i)-(4+6i)
Affix
Irrational Number
four different numbers: i - -i - 1 - and -1.
5. ½(e^(-y) +e^(y)) = cosh y
radicals
complex
cos iy
Complex Division
6. 3rd. Rule of Complex Arithmetic
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
7. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
|z-w|
Rational Number
De Moivre's Theorem
8. I
Imaginary Numbers
Field
v(-1)
How to add and subtract complex numbers (2-3i)-(4+6i)
9. (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
Complex Exponentiation
multiplying complex numbers
cos z
How to multiply complex nubers(2+i)(2i-3)
10. 1
z - z*
i²
How to solve (2i+3)/(9-i)
z1 / z2
11. x + iy = r(cos? + isin?) = re^(i?)
cosh²y - sinh²y
|z-w|
x-axis in the complex plane
Polar Coordinates - z
12. R?¹(cos? - isin?)
Polar Coordinates - z?¹
adding complex numbers
the complex numbers
the distance from z to the origin in the complex plane
13. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
cos iy
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0
14. 3
i^3
|z-w|
complex
Subfield
15. E^(ln r) e^(i?) e^(2pin)
Euler Formula
Absolute Value of a Complex Number
e^(ln z)
real
16. E ^ (z2 ln z1)
Complex Number Formula
Liouville's Theorem -
The Complex Numbers
z1 ^ (z2)
17. 2ib
How to solve (2i+3)/(9-i)
interchangeable
has a solution.
z - z*
18. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
cos z
point of inflection
Integers
Real and Imaginary Parts
19. Imaginary number
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20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
We say that c+di and c-di are complex conjugates.
Absolute Value of a Complex Number
0 if and only if a = b = 0
i^3
21. 1
four different numbers: i - -i - 1 - and -1.
Complex Conjugate
cosh²y - sinh²y
the complex numbers
22. ½(e^(iz) + e^(-iz))
i^2 = -1
standard form of complex numbers
z + z*
cos z
23. A+bi
sin iy
Complex Division
Complex Number Formula
Complex Addition
24. When two complex numbers are added together.
Complex Addition
e^(ln z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number
25. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
z1 ^ (z2)
Imaginary Numbers
Polar Coordinates - z?¹
ln z
26. 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 numbers
multiplying complex numbers
Complex Number
cos iy
27. We can also think of the point z= a+ ib as
the vector (a -b)
Polar Coordinates - Arg(z*)
sin z
Complex Division
28. Not on the numberline
non-integers
Polar Coordinates - Arg(z*)
i^3
(cos? +isin?)n
29. 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
Complex Numbers: Add & subtract
Imaginary Numbers
complex numbers
natural
30. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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31. Where the curvature of the graph changes
(cos? +isin?)n
How to find any Power
Polar Coordinates - Multiplication by i
point of inflection
32. V(x² + y²) = |z|
zz*
Polar Coordinates - r
Argand diagram
Polar Coordinates - z?¹
33. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
For real a and b - a + bi = 0 if and only if a = b = 0
Euler Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two
34. When two complex numbers are multipiled together.
i²
e^(ln z)
Complex Multiplication
subtracting complex numbers
35. R^2 = x
Square Root
integers
i^1
Real Numbers
36. 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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37. A plot of complex numbers as points.
imaginary
Argand diagram
Polar Coordinates - z?¹
Complex Numbers: Multiply
38. When two complex numbers are divided.
Rules of Complex Arithmetic
radicals
Complex Division
a real number: (a + bi)(a - bi) = a² + b²
39. All the powers of i can be written as
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.
i^0
real
40. 1
0 if and only if a = b = 0
Argand diagram
i^4
How to find any Power
41. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
subtracting complex numbers
standard form of complex numbers
Field
The Complex Numbers
42. Given (4-2i) the complex conjugate would be (4+2i)
i^2 = -1
Complex Conjugate
Real and Imaginary Parts
e^(ln z)
43. 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
Rational Number
How to multiply complex nubers(2+i)(2i-3)
subtracting complex numbers
Polar Coordinates - cos?
44. I^2 =
the vector (a -b)
-1
Roots of Unity
conjugate
45. The modulus of the complex number z= a + ib now can be interpreted as
the vector (a -b)
Polar Coordinates - sin?
the distance from z to the origin in the complex plane
complex numbers
46. No i
real
i^2
rational
Every complex number has the 'Standard Form': a + bi for some real a and b.
47. Real and imaginary numbers
complex numbers
'i'
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
48. 2a
multiplying complex numbers
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
49. I = imaginary unit - i² = -1 or i = v-1
Rules of Complex Arithmetic
i^0
natural
Imaginary Numbers
50. 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
has a solution.
Argand diagram
Liouville's Theorem -