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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
the complex numbers
z - z*
standard form of complex numbers
We say that c+di and c-di are complex conjugates.

2. A² + b² - real and non negative
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2 = -1
|z-w|
zz*

3. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Polar Coordinates - z?¹
Subfield
complex numbers

4. All numbers
Polar Coordinates - Multiplication
complex
Polar Coordinates - Arg(z*)
transcendental

5. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
(cos? +isin?)n
sin z
Complex Exponentiation
The Complex Numbers

6. y / r
i²
Square Root
Polar Coordinates - sin?
Polar Coordinates - Division

7. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
The Complex Numbers
four different numbers: i - -i - 1 - and -1.
Absolute Value of a Complex Number
ln z

8. When two complex numbers are subtracted from one another.
Complex Subtraction
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication
Complex Multiplication

9. The square root of -1.
Imaginary Unit
complex numbers
Polar Coordinates - Arg(z*)
Imaginary number

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

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11. Cos n? + i sin n? (for all n integers)
radicals
Imaginary Numbers
'i'
(cos? +isin?)n

12. 3rd. Rule of Complex Arithmetic
Complex Exponentiation
Complex Numbers: Multiply
subtracting complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

13. We can also think of the point z= a+ ib as
the vector (a -b)
Complex numbers are points in the plane
Absolute Value of a Complex Number
x-axis in the complex plane

14. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
For real a and b - a + bi = 0 if and only if a = b = 0
cos iy
has a solution.

15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real
Polar Coordinates - Division

16. I
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
Argand diagram
v(-1)

17. Equivalent to an Imaginary Unit.
|z| = mod(z)
Imaginary number
imaginary
De Moivre's Theorem

18. A + bi
standard form of complex numbers
Complex Number Formula
Complex Exponentiation
complex

19. 1
has a solution.
interchangeable
cosh²y - sinh²y
Liouville's Theorem -

20. 1
i^0
natural
Liouville's Theorem -
subtracting complex numbers

21. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
cos z
zz*
Field

22. 5th. Rule of Complex Arithmetic
the distance from z to the origin in the complex plane
De Moivre's Theorem
conjugate pairs
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. Numbers on a numberline
integers
Absolute Value of a Complex Number
'i'
Real and Imaginary Parts

24. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - Multiplication
Complex Numbers: Multiply
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

25. (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
real
|z| = mod(z)
How to solve (2i+3)/(9-i)
i^4

26. I
Polar Coordinates - Arg(z*)
subtracting complex numbers
Any polynomial O(xn) - (n > 0)
i^1

27. A number that cannot be expressed as a fraction for any integer.
a + bi for some real a and b.
Irrational Number
e^(ln z)
|z| = mod(z)

28. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i²
Roots of Unity
subtracting complex numbers

29. E ^ (z2 ln z1)
z1 ^ (z2)
interchangeable
ln z
Complex Conjugate

30. (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
i^2
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)
a real number: (a + bi)(a - bi) = a² + b²

31. For real a and b - a + bi =
Rules of Complex Arithmetic
Polar Coordinates - Multiplication by i
Complex Number Formula
0 if and only if a = b = 0

32. 2ib
Subfield
z - z*
transcendental
x-axis in the complex plane

33. A complex number may be taken to the power of another complex number.
rational
Complex Exponentiation
Imaginary number
the complex numbers

34. 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.
Affix
sin iy
Complex Number Formula
Complex numbers are points in the plane

35. To simplify the square root of a negative number
real
the distance from z to the origin in the complex plane
How to find any Power
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

36. Derives z = a+bi
How to find any Power
Euler Formula
Complex numbers are points in the plane
subtracting complex numbers

37. The field of all rational and irrational numbers.
Complex Addition
Absolute Value of a Complex Number
Real Numbers
Roots of Unity

38. I^2 =
Integers
Euler Formula
-1
the complex numbers

39. R?¹(cos? - isin?)
Imaginary number
Euler's Formula
i^1
Polar Coordinates - z?¹

40. 2nd. Rule of Complex Arithmetic

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41. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Polar Coordinates - Division
sin iy
real

42. Divide moduli and subtract arguments
imaginary
Polar Coordinates - Division
Real Numbers
Complex Subtraction

43. ? = -tan?
radicals
z1 / z2
Polar Coordinates - Arg(z*)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

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

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45. Have radical
transcendental
irrational
natural
radicals

46. When two complex numbers are divided.
Complex Division
Complex Number Formula
Affix
cos z

47. To simplify a complex fraction
-1
Polar Coordinates - z?¹
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rational Number

48. Has exactly n roots by the fundamental theorem of algebra
i^2
-1
'i'
Any polynomial O(xn) - (n > 0)

49. Written as fractions - terminating + repeating decimals
rational
i^1
|z| = mod(z)
Irrational Number

50. Multiply moduli and add arguments
the complex numbers
a + bi for some real a and b.
conjugate
Polar Coordinates - Multiplication