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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. z1z2* / |z2|²
Polar Coordinates - z
zz*
z1 / z2
Roots of Unity
2. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Division
Complex Number Formula
Polar Coordinates - r
interchangeable
3. 3rd. Rule of Complex Arithmetic
real
z1 / z2
For real a and b - a + bi = 0 if and only if a = b = 0
Irrational Number
4. 1
i^3
Complex numbers are points in the plane
Real Numbers
cosh²y - sinh²y
5. Have radical
radicals
i^0
ln z
complex
6. I = imaginary unit - i² = -1 or i = v-1
The Complex Numbers
Polar Coordinates - z
sin iy
Imaginary Numbers
7. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Argand diagram
transcendental
Imaginary Unit
8. For real a and b - a + bi =
Complex Exponentiation
0 if and only if a = b = 0
Complex Division
Polar Coordinates - r
9. V(zz*) = v(a² + b²)
Subfield
|z| = mod(z)
the vector (a -b)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
10. When two complex numbers are divided.
How to multiply complex nubers(2+i)(2i-3)
sin z
Polar Coordinates - z
Complex Division
11. 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
How to solve (2i+3)/(9-i)
a + bi for some real a and b.
x-axis in the complex plane
Complex Numbers: Add & subtract
12. A plot of complex numbers as points.
Rules of Complex Arithmetic
Argand diagram
i^4
transcendental
13. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)
(cos? +isin?)n
14. ? = -tan?
-1
Complex Subtraction
Polar Coordinates - Arg(z*)
imaginary
15. 1
i²
0 if and only if a = b = 0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural
16. 2ib
the vector (a -b)
How to add and subtract complex numbers (2-3i)-(4+6i)
radicals
z - z*
17. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
The Complex Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)
18. Divide moduli and subtract arguments
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0
rational
Polar Coordinates - Division
19. The field of all rational and irrational numbers.
Real Numbers
Polar Coordinates - z
x-axis in the complex plane
Complex numbers are points in the plane
20. 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
Complex Numbers: Multiply
subtracting complex numbers
zz*
cosh²y - sinh²y
21. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
radicals
Polar Coordinates - z?¹
the complex numbers
22. The product of an imaginary number and its conjugate is
real
ln z
a real number: (a + bi)(a - bi) = a² + b²
How to find any Power
23. Numbers on a numberline
integers
i²
Absolute Value of a Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
24. I^2 =
four different numbers: i - -i - 1 - and -1.
cosh²y - sinh²y
-1
Polar Coordinates - Multiplication
25. Derives z = a+bi
The Complex Numbers
Euler Formula
z1 ^ (z2)
Complex Addition
26. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Polar Coordinates - cos?
conjugate
four different numbers: i - -i - 1 - and -1.
How to add and subtract complex numbers (2-3i)-(4+6i)
27. 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.'
(cos? +isin?)n
subtracting complex numbers
Complex Number
Subfield
28. To simplify the square root of a negative number
Polar Coordinates - Multiplication by i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos iy
cosh²y - sinh²y
29. When two complex numbers are subtracted from one another.
integers
i²
Real and Imaginary Parts
Complex Subtraction
30. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Absolute Value of a Complex Number
Complex Numbers: Add & subtract
point of inflection
31. 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
the vector (a -b)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
32. Starts at 1 - does not include 0
cosh²y - sinh²y
Complex numbers are points in the plane
Complex Number Formula
natural
33. I
Complex Division
|z-w|
v(-1)
complex
34. 1
i^2
conjugate
rational
i^3
35. 3
Polar Coordinates - Arg(z*)
Subfield
Square Root
i^3
36. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
v(-1)
the distance from z to the origin in the complex plane
Imaginary Numbers
37. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
conjugate pairs
Roots of Unity
Complex Numbers: Multiply
z - z*
38. No i
zz*
real
Complex Subtraction
e^(ln z)
39. 1st. Rule of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
integers
i^2 = -1
cosh²y - sinh²y
40. R^2 = x
Square Root
(cos? +isin?)n
irrational
Polar Coordinates - cos?
41. ½(e^(-y) +e^(y)) = cosh y
cos iy
i^2
conjugate pairs
transcendental
42. 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
integers
subtracting complex numbers
the complex numbers
Imaginary number
43. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0
multiplying complex numbers
cos z
44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
zz*
Roots of Unity
z1 ^ (z2)
i^1
45. x / r
Polar Coordinates - cos?
Liouville's Theorem -
Square Root
Complex Numbers: Add & subtract
46. Where the curvature of the graph changes
point of inflection
Square Root
e^(ln z)
rational
47. 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
i^2
Real and Imaginary Parts
Polar Coordinates - Arg(z*)
Euler's Formula
48. When two complex numbers are added together.
Complex Addition
(a + c) + ( b + d)i
Polar Coordinates - Multiplication
cos iy
49. 2nd. Rule of Complex Arithmetic
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50. 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
Square Root
z1 / z2
Rules of Complex Arithmetic
e^(ln z)