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CLEP General Mathematics: Complex Numbers

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. A number that cannot be expressed as a fraction for any integer.
natural
cos z
Irrational Number
Imaginary Numbers

2. Root negative - has letter i
imaginary
Roots of Unity
|z| = mod(z)
Polar Coordinates - sin?

3. Equivalent to an Imaginary Unit.
irrational
Imaginary number
i²
sin iy

4. When two complex numbers are divided.
Field
natural
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Division

5. V(x² + y²) = |z|
Complex Number
Polar Coordinates - r
non-integers
Polar Coordinates - Arg(z*)

6. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
The Complex Numbers
Polar Coordinates - Division
Square Root

7. 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
Affix
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex numbers
the complex numbers

8. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
the complex numbers
Complex Conjugate
cosh²y - sinh²y
ln z

9. When two complex numbers are multipiled together.
natural
We say that c+di and c-di are complex conjugates.
v(-1)
Complex Multiplication

10. 2nd. Rule of Complex Arithmetic

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11. We can also think of the point z= a+ ib as
the vector (a -b)
x-axis in the complex plane
conjugate pairs
sin z

12. I^2 =
-1
i^2
Complex Division
How to add and subtract complex numbers (2-3i)-(4+6i)

13. R?¹(cos? - isin?)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power
Polar Coordinates - z?¹
Real Numbers

14. For real a and b - a + bi =
Argand diagram
0 if and only if a = b = 0
De Moivre's Theorem
standard form of complex numbers

15. Numbers on a numberline
v(-1)
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers

16. 1
How to solve (2i+3)/(9-i)
v(-1)
i²
Polar Coordinates - z

17. I
sin z
(a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
v(-1)

18. The product of an imaginary number and its conjugate is
For real a and b - a + bi = 0 if and only if a = b = 0
zz*
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²

19. E^(ln r) e^(i?) e^(2pin)
For real a and b - a + bi = 0 if and only if a = b = 0
ln z
Complex Exponentiation
e^(ln z)

20. The field of all rational and irrational numbers.
Real Numbers
Polar Coordinates - z?¹
Complex Subtraction
Rules of Complex Arithmetic

21. All the powers of i can be written as
the distance from z to the origin in the complex plane
radicals
i²
four different numbers: i - -i - 1 - and -1.

22. A number that can be expressed as a fraction p/q where q is not equal to 0.
Affix
'i'
Rational Number
standard form of complex numbers

23. No i
real
For real a and b - a + bi = 0 if and only if a = b = 0
cos iy
Any polynomial O(xn) - (n > 0)

24. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Rational Number
the distance from z to the origin in the complex plane
Polar Coordinates - Division

25. 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.
zz*
Complex numbers are points in the plane
Complex Multiplication
x-axis in the complex plane

26. The reals are just the
Euler's Formula
x-axis in the complex plane
Polar Coordinates - Division
Any polynomial O(xn) - (n > 0)

27. ½(e^(-y) +e^(y)) = cosh y
can't get out of the complex numbers by adding (or subtracting) or multiplying two
-1
cos iy
zz*

28. Imaginary number

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29. 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
Real and Imaginary Parts
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)
irrational

30. (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 Numbers: Multiply
four different numbers: i - -i - 1 - and -1.
How to find any Power
How to multiply complex nubers(2+i)(2i-3)

31. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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32. Like pi
transcendental
Roots of Unity
Subfield
-1

33. V(zz*) = v(a² + b²)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z| = mod(z)
multiplying complex numbers
a + bi for some real a and b.

34. Multiply moduli and add arguments
De Moivre's Theorem
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication

35. Rotates anticlockwise by p/2
Complex Number Formula
Any polynomial O(xn) - (n > 0)
(a + c) + ( b + d)i
Polar Coordinates - Multiplication by i

36. xpressions such as ``the complex number z'' - and ``the point z'' are now
sin iy
interchangeable
has a solution.
irrational

37. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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38. z1z2* / |z2|²
Complex numbers are points in the plane
irrational
Real and Imaginary Parts
z1 / z2

39. When two complex numbers are added together.
Complex Addition
i^0
four different numbers: i - -i - 1 - and -1.
For real a and b - a + bi = 0 if and only if a = b = 0

40. 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
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers

41. Derives z = a+bi
i^2
imaginary
Euler Formula
z + z*

42. A plot of complex numbers as points.
Argand diagram
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

43. In this amazing number field every algebraic equation in z with complex coefficients
Polar Coordinates - z
Polar Coordinates - cos?
has a solution.
z + z*

44. 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
natural
Polar Coordinates - cos?
multiplying complex numbers
Rules of Complex Arithmetic

45. Where the curvature of the graph changes
Complex Numbers: Multiply
point of inflection
sin iy
conjugate pairs

46. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
0 if and only if a = b = 0
conjugate
Absolute Value of a Complex Number
imaginary

47. When two complex numbers are subtracted from one another.
has a solution.
a + bi for some real a and b.
standard form of complex numbers
Complex Subtraction

48. 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.
Field
How to find any Power
e^(ln z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

49. 1st. Rule of Complex Arithmetic
i^2 = -1
Every complex number has the 'Standard Form': a + bi for some real a and b.
subtracting complex numbers
Polar Coordinates - cos?

50. A+bi
How to multiply complex nubers(2+i)(2i-3)
i^3
Complex Number Formula
natural