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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. I
De Moivre's Theorem
transcendental
We say that c+di and c-di are complex conjugates.
v(-1)

2. 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.
has a solution.
Complex Number Formula
Complex Numbers: Multiply
Complex Division

3. Starts at 1 - does not include 0
natural
Imaginary Unit
0 if and only if a = b = 0
Argand diagram

4. 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
'i'
Euler Formula
Complex Exponentiation
We say that c+di and c-di are complex conjugates.

5. When two complex numbers are multipiled together.
Rational Number
Complex Multiplication
Real and Imaginary Parts
sin iy

6. The complex number z representing a+bi.
Affix
(cos? +isin?)n
Liouville's Theorem -
Complex Exponentiation

7. 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
How to solve (2i+3)/(9-i)
multiplying complex numbers
z1 ^ (z2)

8. y / r
i^1
Imaginary Unit
Polar Coordinates - sin?
i²

9. 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
Complex Number Formula
a + bi for some real a and b.
Polar Coordinates - z

10. V(zz*) = v(a² + b²)
De Moivre's Theorem
Field
|z| = mod(z)
Absolute Value of a Complex Number

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

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12. Like pi
Real and Imaginary Parts
z - z*
transcendental
non-integers

13. Root negative - has letter i
Complex numbers are points in the plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Add & subtract
imaginary

14. E ^ (z2 ln z1)
Complex Exponentiation
Complex Numbers: Add & subtract
Complex Conjugate
z1 ^ (z2)

15. R?¹(cos? - isin?)
standard form of complex numbers
Polar Coordinates - z?¹
i^3
Polar Coordinates - Division

16. 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.
subtracting complex numbers
Complex numbers are points in the plane
How to find any Power
'i'

17. We can also think of the point z= a+ ib as
Complex numbers are points in the plane
Complex Numbers: Add & subtract
-1
the vector (a -b)

18. 1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
x-axis in the complex plane
i^2
a real number: (a + bi)(a - bi) = a² + b²

19. 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
the complex numbers
natural
For real a and b - a + bi = 0 if and only if a = b = 0
Square Root

20. Have radical
a real number: (a + bi)(a - bi) = a² + b²
radicals
x-axis in the complex plane
point of inflection

21. Real and imaginary numbers
complex numbers
i²
radicals
How to find any Power

22. All the powers of i can be written as
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.
complex numbers
integers

23. 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
point of inflection
Complex Division
Rules of Complex Arithmetic
zz*

24. Has exactly n roots by the fundamental theorem of algebra
z + z*
We say that c+di and c-di are complex conjugates.
Any polynomial O(xn) - (n > 0)
Complex Number

25. The reals are just the
Roots of Unity
adding complex numbers
x-axis in the complex plane
has a solution.

26. Rotates anticlockwise by p/2
Rational Number
non-integers
Polar Coordinates - Multiplication by i
For real a and b - a + bi = 0 if and only if a = b = 0

27. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Polar Coordinates - Multiplication
Imaginary Unit
Real Numbers

28. The square root of -1.
Polar Coordinates - r
Imaginary Unit
Complex numbers are points in the plane
has a solution.

29. x / r
Rules of Complex Arithmetic
Polar Coordinates - cos?
Polar Coordinates - Division
i^3

30. Not on the numberline
Complex Numbers: Add & subtract
Affix
non-integers
conjugate pairs

31. E^(ln r) e^(i?) e^(2pin)
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z-w|
cosh²y - sinh²y
e^(ln z)

32. A + bi
Polar Coordinates - Arg(z*)
integers
standard form of complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

33. When two complex numbers are divided.
Complex Division
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²
natural

34. (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
Subfield
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - cos?
(a + c) + ( b + d)i

35. A complex number may be taken to the power of another complex number.
Polar Coordinates - z?¹
z - z*
Polar Coordinates - r
Complex Exponentiation

36. 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
Euler Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Square Root
subtracting complex numbers

37. Divide moduli and subtract arguments
Complex Division
rational
Complex Exponentiation
Polar Coordinates - Division

38. Imaginary number

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39. To simplify the square root of a negative number
multiplying complex numbers
radicals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Integers

40. I
Polar Coordinates - z?¹
i^1
natural
subtracting complex numbers

41. Multiply moduli and add arguments
z1 ^ (z2)
i^3
Polar Coordinates - Multiplication
Roots of Unity

42. A+bi
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)
Complex Subtraction
Complex Number Formula

43. No i
zz*
real
Polar Coordinates - sin?
adding complex numbers

44. A plot of complex numbers as points.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
non-integers
Rules of Complex Arithmetic
Argand diagram

45. A² + b² - real and non negative
Subfield
Polar Coordinates - Multiplication
'i'
zz*

46. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Division
i^2
ln z
|z-w|

47. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
Polar Coordinates - Multiplication
Polar Coordinates - Arg(z*)
a real number: (a + bi)(a - bi) = a² + b²

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.
z + z*
Roots of Unity
conjugate
Complex numbers are points in the plane

49. Written as fractions - terminating + repeating decimals
(a + c) + ( b + d)i
the complex numbers
rational
(cos? +isin?)n

50. When two complex numbers are added together.
Complex Addition
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
the complex numbers