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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. To simplify the square root of a negative number
i²
i^2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
0 if and only if a = b = 0

2. 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
For real a and b - a + bi = 0 if and only if a = b = 0
Real and Imaginary Parts
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²

3. 1
Complex Number
'i'
i^0
Argand diagram

4. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - r
irrational
Irrational Number
i^2

5. 1
i²
Complex Addition
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.

6. 1
z - z*
i^4
-1
real

7. (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)
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

8. z1z2* / |z2|²
z1 / z2
the distance from z to the origin in the complex plane
Euler Formula
transcendental

9. (e^(-y) - e^(y)) / 2i = i sinh y
Argand diagram
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin iy

10. A plot of complex numbers as points.
imaginary
Argand diagram
Liouville's Theorem -
z1 / z2

11. Where the curvature of the graph changes
(a + c) + ( b + d)i
point of inflection
Polar Coordinates - Division
subtracting complex numbers

12. 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
Subfield
z1 / z2
How to solve (2i+3)/(9-i)

13. (a + bi) = (c + bi) =
Complex Number Formula
(a + c) + ( b + d)i
the complex numbers
standard form of complex numbers

14. R?¹(cos? - isin?)
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)
Real and Imaginary Parts
Polar Coordinates - z?¹

15. A² + b² - real and non negative
ln z
The Complex Numbers
zz*
cos z

16. 2ib
real
|z-w|
z - z*
radicals

17. All the powers of i can be written as
Irrational Number
four different numbers: i - -i - 1 - and -1.
conjugate pairs
Subfield

18. 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
(cos? +isin?)n
0 if and only if a = b = 0
Complex Numbers: Multiply

19. 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
0 if and only if a = b = 0
ln z
Field
subtracting complex numbers

20. 3rd. Rule of Complex Arithmetic
Complex Number
z - z*
Polar Coordinates - Division
For real a and b - a + bi = 0 if and only if a = b = 0

21. 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.
Polar Coordinates - z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
sin z
Complex Numbers: Multiply

22. In this amazing number field every algebraic equation in z with complex coefficients
(a + bi) = (c + bi) = (a + c) + ( b + d)i
'i'
has a solution.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. A subset within a field.
complex numbers
Subfield
interchangeable
Polar Coordinates - Arg(z*)

24. A complex number and its conjugate
Imaginary Unit
Imaginary Numbers
conjugate pairs
Polar Coordinates - Multiplication by i

25. Have radical
Imaginary number
complex
radicals
Polar Coordinates - Multiplication by i

26. y / r
subtracting complex numbers
radicals
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?

27. Like pi
the complex numbers
z - z*
Complex Number
transcendental

28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Subfield
Argand diagram
non-integers

29. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Exponentiation
cos iy
The Complex Numbers
Irrational Number

30. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
complex numbers
multiplying complex numbers
interchangeable
Argand diagram

31. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
z1 ^ (z2)
Complex Multiplication
Field

32. Written as fractions - terminating + repeating decimals
natural
transcendental
the complex numbers
rational

33. A+bi
Complex Number Formula
Polar Coordinates - sin?
has a solution.
adding complex numbers

34. When two complex numbers are added together.
Rules of Complex Arithmetic
Complex Addition
Real Numbers
Irrational Number

35. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
Liouville's Theorem -
Complex Number
Polar Coordinates - z

36. 4th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
Imaginary Numbers
Polar Coordinates - z
(a + bi) = (c + bi) = (a + c) + ( b + d)i

37. The reals are just the
x-axis in the complex plane
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Any polynomial O(xn) - (n > 0)

38. Imaginary number

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39. (a + bi)(c + bi) =
How to solve (2i+3)/(9-i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
rational
conjugate

40. When two complex numbers are divided.
has a solution.
Polar Coordinates - r
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Division

41. Cos n? + i sin n? (for all n integers)
How to find any Power
standard form of complex numbers
Imaginary Numbers
(cos? +isin?)n

42. We can also think of the point z= a+ ib as
sin iy
the vector (a -b)
radicals
rational

43. Not on the numberline
Absolute Value of a Complex Number
Polar Coordinates - Division
non-integers
De Moivre's Theorem

44. Every complex number has the 'Standard Form':
the complex numbers
a + bi for some real a and b.
-1
Real and Imaginary Parts

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

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46. 1
Complex Exponentiation
Argand diagram
a + bi for some real a and b.
cosh²y - sinh²y

47. R^2 = x
rational
Complex Division
Square Root
a + bi for some real a and b.

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.
x-axis in the complex plane
How to find any Power
The Complex Numbers
standard form of complex numbers

49. A complex number may be taken to the power of another complex number.
How to multiply complex nubers(2+i)(2i-3)
conjugate
i^2
Complex Exponentiation

50. When two complex numbers are multipiled together.
Complex Multiplication
'i'
z1 / z2
cos iy