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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. No i
i^4
real
sin z
non-integers

2. Starts at 1 - does not include 0
sin z
the distance from z to the origin in the complex plane
natural
Field

3. 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.
|z| = mod(z)
i²
cos z
Complex Numbers: Multiply

4. 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
Complex Conjugate
the complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts

5. 2nd. Rule of Complex Arithmetic

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6. 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.
Complex Division
adding complex numbers
How to find any Power
Affix

7. Derives z = a+bi
Complex numbers are points in the plane
Euler Formula
i^0
four different numbers: i - -i - 1 - and -1.

8. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Affix
conjugate
i²
z - z*

9. Divide moduli and subtract arguments
How to multiply complex nubers(2+i)(2i-3)
Complex Subtraction
Complex Exponentiation
Polar Coordinates - Division

10. I
Real and Imaginary Parts
rational
i^1
multiplying complex numbers

11. A number that cannot be expressed as a fraction for any integer.
radicals
Irrational Number
The Complex Numbers
conjugate pairs

12. 1
subtracting complex numbers
i²
Polar Coordinates - Arg(z*)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

13. Numbers on a numberline
We say that c+di and c-di are complex conjugates.
integers
How to find any Power
Polar Coordinates - Arg(z*)

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

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15. All the powers of i can be written as
|z-w|
Polar Coordinates - Multiplication
four different numbers: i - -i - 1 - and -1.
conjugate

16. Cos n? + i sin n? (for all n integers)
i^0
irrational
(cos? +isin?)n
Euler's Formula

17. Like pi
Polar Coordinates - z
transcendental
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.

18. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
i^3
Polar Coordinates - cos?
Complex Number

19. To simplify the square root of a negative number
multiplying complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Irrational Number

20. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Liouville's Theorem -
Euler's Formula
Subfield

21. A complex number may be taken to the power of another complex number.
Complex numbers are points in the plane
Complex Exponentiation
adding complex numbers
Euler Formula

22. V(x² + y²) = |z|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin iy
We say that c+di and c-di are complex conjugates.
Polar Coordinates - r

23. (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
How to solve (2i+3)/(9-i)
i^2
Real and Imaginary Parts
Rational Number

24. 3
0 if and only if a = b = 0
imaginary
i^3
Polar Coordinates - Multiplication by i

25. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Imaginary Numbers
multiplying complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field

26. E ^ (z2 ln z1)
i^4
Imaginary Numbers
z1 ^ (z2)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. 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
For real a and b - a + bi = 0 if and only if a = b = 0
zz*
How to multiply complex nubers(2+i)(2i-3)
We say that c+di and c-di are complex conjugates.

28. Multiply moduli and add arguments
multiplying complex numbers
Polar Coordinates - Multiplication
Complex Multiplication
Complex Number Formula

29. When two complex numbers are added together.
ln z
i²
Complex Addition
For real a and b - a + bi = 0 if and only if a = b = 0

30. R^2 = x
i^2 = -1
Square Root
i²
Roots of Unity

31. x / r
Polar Coordinates - cos?
the vector (a -b)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
'i'

32. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Arg(z*)
Real and Imaginary Parts
Imaginary Unit
Rational Number

33. 1
i^0
transcendental
the vector (a -b)
imaginary

34. Real and imaginary numbers
Imaginary Unit
rational
Roots of Unity
complex numbers

35. A² + b² - real and non negative
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*
irrational
sin iy

36. A + bi
imaginary
De Moivre's Theorem
standard form of complex numbers
i^1

37. All numbers
How to multiply complex nubers(2+i)(2i-3)
complex
The Complex Numbers
non-integers

38. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Number
Affix
i^2

39. For real a and b - a + bi =
z1 / z2
Real Numbers
natural
0 if and only if a = b = 0

40. Given (4-2i) the complex conjugate would be (4+2i)
Subfield
Complex Conjugate
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

41. In this amazing number field every algebraic equation in z with complex coefficients
subtracting complex numbers
the vector (a -b)
real
has a solution.

42. 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
Imaginary number
conjugate pairs
(cos? +isin?)n
Rules of Complex Arithmetic

43. 2a
i^1
z + z*
How to multiply complex nubers(2+i)(2i-3)
i^4

44. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
ln z
cos z
conjugate pairs

45. Root negative - has letter i
imaginary
conjugate
De Moivre's Theorem
v(-1)

46. Has exactly n roots by the fundamental theorem of algebra
Roots of Unity
Complex Exponentiation
Any polynomial O(xn) - (n > 0)
sin iy

47. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Complex numbers are points in the plane
non-integers
x-axis in the complex plane

48. 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.
sin z
real
Complex numbers are points in the plane
v(-1)

49. A plot of complex numbers as points.
Real Numbers
Polar Coordinates - z
Argand diagram
cos iy

50. 1st. Rule of Complex Arithmetic
e^(ln z)
Rational Number
Polar Coordinates - cos?
i^2 = -1