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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. 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
Imaginary Unit
natural
i^3

2. E^(ln r) e^(i?) e^(2pin)
i^3
e^(ln z)
Polar Coordinates - cos?
Complex Number Formula

3. 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.
a real number: (a + bi)(a - bi) = a² + b²
How to find any Power
integers
subtracting complex numbers

4. The reals are just the
irrational
Integers
|z-w|
x-axis in the complex plane

5. Every complex number has the 'Standard Form':
Real and Imaginary Parts
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a + bi for some real a and b.
Integers

6. 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
(cos? +isin?)n
subtracting complex numbers
Rational Number
Polar Coordinates - Multiplication by i

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
How to multiply complex nubers(2+i)(2i-3)
transcendental
0 if and only if a = b = 0

8. All numbers
complex
Complex Conjugate
Euler Formula
Polar Coordinates - z

9. Where the curvature of the graph changes
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
point of inflection

10. Numbers on a numberline
four different numbers: i - -i - 1 - and -1.
integers
Complex Numbers: Multiply
Complex Multiplication

11. ½(e^(-y) +e^(y)) = cosh y
Absolute Value of a Complex Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram
cos iy

12. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Conjugate

13. For real a and b - a + bi =
0 if and only if a = b = 0
Complex Number
real
has a solution.

14. Cos n? + i sin n? (for all n integers)
Complex Numbers: Add & subtract
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.
We say that c+di and c-di are complex conjugates.

15. 5th. Rule of Complex Arithmetic
Integers
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers

16. 1
How to multiply complex nubers(2+i)(2i-3)
z - z*
rational
i^0

17. A plot of complex numbers as points.
Argand diagram
Imaginary Numbers
Complex Numbers: Add & subtract
integers

18. When two complex numbers are divided.
Complex Division
imaginary
transcendental
the distance from z to the origin in the complex plane

19. x / r
can't get out of the complex numbers by adding (or subtracting) or multiplying two
radicals
rational
Polar Coordinates - cos?

20. 1
i^2
i^2 = -1
four different numbers: i - -i - 1 - and -1.
cosh²y - sinh²y

21. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Roots of Unity
i^2
conjugate
We say that c+di and c-di are complex conjugates.

22. (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
standard form of complex numbers
multiplying complex numbers
i^2 = -1
How to multiply complex nubers(2+i)(2i-3)

23. 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)
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|

24. (e^(iz) - e^(-iz)) / 2i
sin z
interchangeable
0 if and only if a = b = 0
v(-1)

25. V(x² + y²) = |z|
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex numbers are points in the plane

26. Any number not rational
conjugate
De Moivre's Theorem
cos z
irrational

27. The complex number z representing a+bi.
Affix
Polar Coordinates - Multiplication by i
Polar Coordinates - cos?
i^1

28. y / r
De Moivre's Theorem
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n

29. A + bi
i^2 = -1
Imaginary Numbers
standard form of complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

30. Derives z = a+bi
transcendental
standard form of complex numbers
Euler Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i

31. Written as fractions - terminating + repeating decimals
rational
Rational Number
cos z
v(-1)

32. 3
Polar Coordinates - Multiplication
i^3
-1
How to multiply complex nubers(2+i)(2i-3)

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

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34. 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
Affix
Rules of Complex Arithmetic
Liouville's Theorem -
Imaginary number

35. 2nd. Rule of Complex Arithmetic

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36. Divide moduli and subtract arguments
conjugate pairs
a + bi for some real a and b.
the vector (a -b)
Polar Coordinates - Division

37. E ^ (z2 ln z1)
z1 ^ (z2)
radicals
Polar Coordinates - cos?
Real Numbers

38. R?¹(cos? - isin?)
a + bi for some real a and b.
Polar Coordinates - z?¹
i^2 = -1
Real and Imaginary Parts

39. (a + bi) = (c + bi) =
Complex Division
Complex Multiplication
z1 / z2
(a + c) + ( b + d)i

40. In this amazing number field every algebraic equation in z with complex coefficients
Complex Addition
the distance from z to the origin in the complex plane
has a solution.
cos iy

41. The modulus of the complex number z= a + ib now can be interpreted as
i^3
the distance from z to the origin in the complex plane
Rational Number
Imaginary Unit

42. 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
Liouville's Theorem -
Subfield
i^2
Real and Imaginary Parts

43. No i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^4
Imaginary Numbers
real

44. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
conjugate pairs
Affix
Integers
the complex numbers

45. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
z - z*
Square Root
point of inflection

46. The field of all rational and irrational numbers.
Polar Coordinates - cos?
Polar Coordinates - r
Real Numbers
z1 / z2

47. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin iy
i^4

48. A subset within a field.
Complex Numbers: Multiply
multiplying complex numbers
Argand diagram
Subfield

49. Like pi
transcendental
Complex Subtraction
zz*
imaginary

50. 2ib
z - z*
standard form of complex numbers
non-integers
cos z