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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. 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 Exponentiation
the vector (a -b)
Imaginary number
Real and Imaginary Parts

2. 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
Euler's Formula
i^0
Complex Addition
Complex Numbers: Add & subtract

3. 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
subtracting complex numbers
Argand diagram
complex numbers
Complex Number

4. Like pi
the complex numbers
Complex Division
transcendental
De Moivre's Theorem

5. 1
Polar Coordinates - Multiplication by i
i^2
De Moivre's Theorem
Rational Number

6. 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)
Every complex number has the 'Standard Form': a + bi for some real a and b.
multiply the numerator and the denominator by the complex conjugate of the denominator.
v(-1)

7. The product of an imaginary number and its conjugate is
The Complex Numbers
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z
has a solution.

8. A complex number and its conjugate
real
v(-1)
conjugate pairs
i^0

9. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
ln z
Absolute Value of a Complex Number
Complex Numbers: Add & subtract
z1 ^ (z2)

10. The reals are just the
cos z
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane

11. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
x-axis in the complex plane
De Moivre's Theorem
Polar Coordinates - z?¹

12. When two complex numbers are divided.
Complex Multiplication
Polar Coordinates - Arg(z*)
Complex Division
0 if and only if a = b = 0

13. A plot of complex numbers as points.
|z-w|
Argand diagram
Polar Coordinates - Multiplication
radicals

14. Derives z = a+bi
i^0
i^2
Euler Formula
Imaginary Numbers

15. I
v(-1)
i^2 = -1
Complex Numbers: Multiply
z - z*

16. Imaginary number

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17. A number that can be expressed as a fraction p/q where q is not equal to 0.
conjugate pairs
Complex Number
Rational Number
Argand diagram

18. 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.'
|z-w|
non-integers
irrational
Complex Number

19. 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.
Complex numbers are points in the plane
interchangeable
z + z*
Euler Formula

20. Any number not rational
Rational Number
interchangeable
i²
irrational

21. ? = -tan?
Polar Coordinates - Arg(z*)
ln z
The Complex Numbers
Field

22. Has exactly n roots by the fundamental theorem of algebra
How to find any Power
z1 ^ (z2)
Complex numbers are points in the plane
Any polynomial O(xn) - (n > 0)

23. All numbers
Imaginary Unit
complex
Any polynomial O(xn) - (n > 0)
i²

24. 2ib
Square Root
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z - z*
Real Numbers

25. x + iy = r(cos? + isin?) = re^(i?)
sin iy
(a + c) + ( b + d)i
Polar Coordinates - z
Polar Coordinates - Multiplication

26. 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.
i^1
Integers
Complex Numbers: Multiply
i^2

27. Where the curvature of the graph changes
Complex Numbers: Add & subtract
imaginary
point of inflection
(cos? +isin?)n

28. (e^(-y) - e^(y)) / 2i = i sinh y
real
0 if and only if a = b = 0
sin iy
conjugate

29. A² + b² - real and non negative
Rules of Complex Arithmetic
complex numbers
zz*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. 1
e^(ln z)
Polar Coordinates - z
i^4
Imaginary Unit

31. No i
real
De Moivre's Theorem
i^1
-1

32. 3
Complex Number Formula
i^3
Complex Multiplication
integers

33. 1st. Rule of Complex Arithmetic
We say that c+di and c-di are complex conjugates.
Irrational Number
transcendental
i^2 = -1

34. R?¹(cos? - isin?)
(cos? +isin?)n
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - z?¹
Polar Coordinates - Division

35. 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
integers
How to find any Power
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z?¹

36. 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
Rules of Complex Arithmetic
0 if and only if a = b = 0
Imaginary number
i²

37. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Euler Formula
Integers
transcendental
i^1

38. Equivalent to an Imaginary Unit.
sin z
Imaginary number
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. We see in this way that the distance between two points z and w in the complex plane is
i^3
Polar Coordinates - Division
Polar Coordinates - cos?
|z-w|

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

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41. ½(e^(iz) + e^(-iz))
x-axis in the complex plane
cos z
Polar Coordinates - Multiplication
Euler Formula

42. 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 Number Formula
Any polynomial O(xn) - (n > 0)
i^0
How to find any Power

43. 2a
cosh²y - sinh²y
Polar Coordinates - Multiplication by i
z + z*
cos z

44. Given (4-2i) the complex conjugate would be (4+2i)
i²
Complex Conjugate
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.

45. In this amazing number field every algebraic equation in z with complex coefficients
Polar Coordinates - Multiplication
has a solution.
-1
Polar Coordinates - z

46. The field of all rational and irrational numbers.
Irrational Number
Real Numbers
Polar Coordinates - Multiplication by i
i^0

47. When two complex numbers are subtracted from one another.
Irrational Number
Real Numbers
Complex Subtraction
|z| = mod(z)

48. V(x² + y²) = |z|
Polar Coordinates - r
The Complex Numbers
Polar Coordinates - sin?
i^3

49. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
irrational
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0

50. I^2 =
Absolute Value of a Complex Number
Complex Number
-1
Imaginary Unit