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CLEP General Mathematics: Complex Numbers

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. Any number not rational
irrational
complex
Subfield
i^3

2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
ln z
De Moivre's Theorem
The Complex Numbers
Field

3. ? = -tan?
Polar Coordinates - Arg(z*)
adding complex numbers
v(-1)
Roots of Unity

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

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5. 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
complex numbers
Polar Coordinates - sin?
Real Numbers
adding complex numbers

6. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - sin?

7. x + iy = r(cos? + isin?) = re^(i?)
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z
four different numbers: i - -i - 1 - and -1.

8. 1
i^4
transcendental
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. To simplify the square root of a negative number
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
De Moivre's Theorem
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

10. The product of an imaginary number and its conjugate is
transcendental
We say that c+di and c-di are complex conjugates.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a real number: (a + bi)(a - bi) = a² + b²

11. z1z2* / |z2|²
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.
z1 / z2

12. I^2 =
-1
Complex Division
Complex Number Formula
interchangeable

13. Real and imaginary numbers
real
complex numbers
Complex Number Formula
radicals

14. All the powers of i can be written as
De Moivre's Theorem
irrational
four different numbers: i - -i - 1 - and -1.
multiply the numerator and the denominator by the complex conjugate of the denominator.

15. The field of all rational and irrational numbers.
Real Numbers
the distance from z to the origin in the complex plane
Complex Numbers: Add & subtract
four different numbers: i - -i - 1 - and -1.

16. 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.
has a solution.
How to find any Power
(cos? +isin?)n
z1 ^ (z2)

17. E ^ (z2 ln z1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cosh²y - sinh²y
Euler Formula
z1 ^ (z2)

18. A complex number and its conjugate
i^4
Complex Addition
conjugate pairs
a + bi for some real a and b.

19. Have radical
radicals
cos z
complex numbers
complex

20. 1
sin z
Imaginary Unit
Polar Coordinates - Arg(z*)
i²

21. 1
z + z*
cosh²y - sinh²y
a real number: (a + bi)(a - bi) = a² + b²
sin iy

22. When two complex numbers are added together.
Real and Imaginary Parts
The Complex Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Addition

23. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - Arg(z*)
sin iy
Polar Coordinates - cos?
point of inflection

24. (a + bi)(c + bi) =
Polar Coordinates - r
-1
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

25. (e^(iz) - e^(-iz)) / 2i
0 if and only if a = b = 0
Polar Coordinates - z
How to multiply complex nubers(2+i)(2i-3)
sin z

26. 1
How to multiply complex nubers(2+i)(2i-3)
Rational Number
i^0
Absolute Value of a Complex Number

27. V(x² + y²) = |z|
Liouville's Theorem -
z1 ^ (z2)
Polar Coordinates - r
imaginary

28. 4th. Rule of Complex Arithmetic
Any polynomial O(xn) - (n > 0)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin iy
Complex Numbers: Add & subtract

29. 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
(a + c) + ( b + d)i
imaginary
Imaginary Unit
subtracting complex numbers

30. For real a and b - a + bi =
Field
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
(cos? +isin?)n

31. A + bi
non-integers
i²
How to solve (2i+3)/(9-i)
standard form of complex numbers

32. I
v(-1)
a + bi for some real a and b.
|z| = mod(z)
Square Root

33. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
|z| = mod(z)
Complex numbers are points in the plane
a + bi for some real a and b.

34. A number that cannot be expressed as a fraction for any integer.
Irrational Number
four different numbers: i - -i - 1 - and -1.
Euler Formula
the vector (a -b)

35. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^2
How to find any Power
multiplying complex numbers
Complex Number

36. All numbers
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
natural
the vector (a -b)

37. 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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38. R^2 = x
How to add and subtract complex numbers (2-3i)-(4+6i)
Any polynomial O(xn) - (n > 0)
Square Root
i^1

39. Derives z = a+bi
Complex Subtraction
the vector (a -b)
Euler Formula
Imaginary Numbers

40. Numbers on a numberline
integers
z1 ^ (z2)
Irrational Number
cosh²y - sinh²y

41. 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.
Complex Numbers: Multiply
transcendental
zz*
|z-w|

42. Has exactly n roots by the fundamental theorem of algebra
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
v(-1)
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r

43. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
standard form of complex numbers
Complex Number Formula
natural

44. ½(e^(iz) + e^(-iz))
x-axis in the complex plane
natural
cos z
Complex Addition

45. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Division
Real and Imaginary Parts
Rules of Complex Arithmetic
e^(ln z)

46. 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.'
Real and Imaginary Parts
Polar Coordinates - Multiplication by i
i^4
Complex Number

47. No i
z1 ^ (z2)
i^2
real
Complex Subtraction

48. When two complex numbers are multipiled together.
Complex Multiplication
i^0
imaginary
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

49. When two complex numbers are divided.
v(-1)
Complex Division
cosh²y - sinh²y
Complex Numbers: Add & subtract

50. Like pi
i^4
a + bi for some real a and b.
'i'
transcendental