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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. 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)
conjugate pairs
integers
cosh²y - sinh²y

2. Root negative - has letter i
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a real number: (a + bi)(a - bi) = a² + b²
imaginary

3. (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)
sin iy
Rational Number
real

4. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - r
Field
Complex Exponentiation
How to find any Power

5. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
v(-1)
ln z
Irrational Number
real

6. A + bi
Square Root
i^0
standard form of complex numbers
the distance from z to the origin in the complex plane

7. A number that can be expressed as a fraction p/q where q is not equal to 0.
has a solution.
Complex Multiplication
Polar Coordinates - sin?
Rational Number

8. The reals are just the
Subfield
x-axis in the complex plane
point of inflection
Real and Imaginary Parts

9. When two complex numbers are multipiled together.
Complex Multiplication
sin z
How to find any Power
the vector (a -b)

10. I^2 =
-1
Polar Coordinates - Arg(z*)
imaginary
How to find any Power

11. 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
i^2
Complex Exponentiation
Rules of Complex Arithmetic
the distance from z to the origin in the complex plane

12. Derives z = a+bi
Euler Formula
How to multiply complex nubers(2+i)(2i-3)
Absolute Value of a Complex Number
natural

13. Every complex number has the 'Standard Form':
i^2
Imaginary Unit
a + bi for some real a and b.
Polar Coordinates - r

14. When two complex numbers are divided.
z - z*
Complex Division
Square Root
multiply the numerator and the denominator by the complex conjugate of the denominator.

15. 3
imaginary
i^3
(cos? +isin?)n
adding complex numbers

16. E^(ln r) e^(i?) e^(2pin)
z1 ^ (z2)
e^(ln z)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula

17. All the powers of i can be written as
zz*
Polar Coordinates - r
Square Root
four different numbers: i - -i - 1 - and -1.

18. Like pi
Rules of Complex Arithmetic
De Moivre's Theorem
Polar Coordinates - sin?
transcendental

19. 1
complex numbers
i^2
(cos? +isin?)n
Polar Coordinates - sin?

20. I
v(-1)
Roots of Unity
z - z*
ln z

21. A subset within a field.
e^(ln z)
Complex Exponentiation
Subfield
four different numbers: i - -i - 1 - and -1.

22. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
Integers
The Complex Numbers
the vector (a -b)
How to solve (2i+3)/(9-i)

23. Cos n? + i sin n? (for all n integers)
imaginary
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.
Euler Formula

24. A number that cannot be expressed as a fraction for any integer.
Irrational Number
|z| = mod(z)
Euler Formula
cosh²y - sinh²y

25. 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
adding complex numbers
-1
radicals
integers

26. I
Real Numbers
Polar Coordinates - Division
How to solve (2i+3)/(9-i)
i^1

27. 1
i^4
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication

28. The product of an imaginary number and its conjugate is
cos iy
Absolute Value of a Complex Number
i^2 = -1
a real number: (a + bi)(a - bi) = a² + b²

29. The square root of -1.
Field
Real Numbers
Irrational Number
Imaginary Unit

30. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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31. 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
Complex Addition
multiplying complex numbers

32. 2ib
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
z - z*

33. For real a and b - a + bi =
The Complex Numbers
0 if and only if a = b = 0
De Moivre's Theorem
Complex Division

34. A complex number may be taken to the power of another complex number.
Complex Exponentiation
sin iy
sin z
Any polynomial O(xn) - (n > 0)

35. (e^(iz) - e^(-iz)) / 2i
i^2 = -1
sin z
Field
transcendental

36. 1
i²
(cos? +isin?)n
standard form of complex numbers
irrational

37. 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.
the vector (a -b)
Complex numbers are points in the plane
irrational
-1

38. In this amazing number field every algebraic equation in z with complex coefficients
Complex Number
has a solution.
transcendental
Polar Coordinates - Division

39. A complex number and its conjugate
adding complex numbers
How to solve (2i+3)/(9-i)
conjugate pairs
imaginary

40. When two complex numbers are added together.
Imaginary number
Complex Addition
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power

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

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42. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number Formula
Irrational Number

43. 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.
We say that c+di and c-di are complex conjugates.
The Complex Numbers
Complex Numbers: Multiply
Complex Conjugate

44. A+bi
Integers
Polar Coordinates - r
Complex Number Formula
has a solution.

45. 1
Complex Conjugate
cosh²y - sinh²y
adding complex numbers
Complex Multiplication

46. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
Subfield
irrational
complex

47. When two complex numbers are subtracted from one another.
has a solution.
i^2 = -1
Imaginary number
Complex Subtraction

48. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit
subtracting complex numbers
Real and Imaginary Parts

49. Any number not rational
irrational
cos iy
has a solution.
How to find any Power

50. x / r
irrational
Complex Addition
the complex numbers
Polar Coordinates - cos?