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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.
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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. 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
z1 ^ (z2)
natural
the complex numbers
Imaginary Numbers

2. Imaginary number

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3. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Polar Coordinates - sin?
Absolute Value of a Complex Number
transcendental

4. A² + b² - real and non negative
Polar Coordinates - z
v(-1)
zz*
Subfield

5. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
|z-w|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs

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

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7. 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
Complex Number Formula
irrational
We say that c+di and c-di are complex conjugates.
the distance from z to the origin in the complex plane

8. For real a and b - a + bi =
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to solve (2i+3)/(9-i)
0 if and only if a = b = 0
Irrational Number

9. Numbers on a numberline
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)
integers
a real number: (a + bi)(a - bi) = a² + b²

10. 3rd. Rule of Complex Arithmetic
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0
zz*
Complex Numbers: Add & subtract

11. I
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division
v(-1)
How to solve (2i+3)/(9-i)

12. Multiply moduli and add arguments
Polar Coordinates - Multiplication
cos z
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

13. All the powers of i can be written as
Roots of Unity
the complex numbers
zz*
four different numbers: i - -i - 1 - and -1.

14. 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.
rational
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply
sin iy

15. ½(e^(-y) +e^(y)) = cosh y
cos iy
irrational
subtracting complex numbers
Field

16. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiplying complex numbers
Rules of Complex Arithmetic

17. Equivalent to an Imaginary Unit.
Integers
Imaginary number
complex
Absolute Value of a Complex Number

18. A complex number and its conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
(cos? +isin?)n
i^1

19. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - Multiplication by i
i^0
Complex Conjugate
multiplying complex numbers

20. Every complex number has the 'Standard Form':
a + bi for some real a and b.
interchangeable
real
natural

21. (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
The Complex Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
z1 ^ (z2)

22. 3
i^3
integers
cos iy
non-integers

23. 5th. Rule of Complex Arithmetic
point of inflection
the complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z

24. All numbers
Polar Coordinates - Division
imaginary
complex
How to multiply complex nubers(2+i)(2i-3)

25. A + bi
z1 ^ (z2)
Complex Number Formula
standard form of complex numbers
Integers

26. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Complex Numbers: Add & subtract
Real and Imaginary Parts
i^2

27. Not on the numberline
non-integers
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
natural

28. The field of all rational and irrational numbers.
i^3
Complex Addition
i^0
Real Numbers

29. To simplify the square root of a negative number
Euler's Formula
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i²

30. To simplify a complex fraction
Complex Conjugate
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler's Formula

31. A number that cannot be expressed as a fraction for any integer.
(cos? +isin?)n
Irrational Number
complex
ln z

32. 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
x-axis in the complex plane
Complex Numbers: Add & subtract
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex

33. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^4
point of inflection
Rational Number
Integers

34. (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)
Euler Formula
Field
Polar Coordinates - z

35. Given (4-2i) the complex conjugate would be (4+2i)
Complex Exponentiation
Argand diagram
Integers
Complex Conjugate

36. Derives z = a+bi
Euler Formula
Polar Coordinates - z?¹
Argand diagram
Field

37. When two complex numbers are divided.
z - z*
Integers
Complex Division
Polar Coordinates - Arg(z*)

38. (e^(iz) - e^(-iz)) / 2i
rational
De Moivre's Theorem
Polar Coordinates - sin?
sin z

39. (a + bi)(c + bi) =
Rules of Complex Arithmetic
Polar Coordinates - Division
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

40. Any number not rational
(a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
irrational
ln z

41. 1
Polar Coordinates - r
Roots of Unity
Absolute Value of a Complex Number
i^4

42. We can also think of the point z= a+ ib as
Complex Numbers: Multiply
the vector (a -b)
Complex Multiplication
Integers

43. The reals are just the
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to solve (2i+3)/(9-i)
Complex Numbers: Add & subtract

44. 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
Complex Conjugate
interchangeable
Real and Imaginary Parts
e^(ln z)

45. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - sin?
zz*
Complex Number
Any polynomial O(xn) - (n > 0)

46. E ^ (z2 ln z1)
i^2
z1 ^ (z2)
Affix
Polar Coordinates - cos?

47. I = imaginary unit - i² = -1 or i = v-1
irrational
Argand diagram
Imaginary Numbers
Complex Conjugate

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.
How to find any Power
transcendental
Polar Coordinates - Multiplication by i
the complex numbers

49. Divide moduli and subtract arguments
Real Numbers
Polar Coordinates - Division
Polar Coordinates - Multiplication
complex numbers

50. E^(ln r) e^(i?) e^(2pin)
non-integers
e^(ln z)
De Moivre's Theorem
Imaginary Numbers