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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. 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
Imaginary Unit
integers
Liouville's Theorem -

2. 5th. Rule of Complex Arithmetic
real
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)
Imaginary Unit

3. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Division
interchangeable
For real a and b - a + bi = 0 if and only if a = b = 0
irrational

4. Written as fractions - terminating + repeating decimals
rational
zz*
z + z*
conjugate

5. R?¹(cos? - isin?)
interchangeable
Polar Coordinates - z?¹
non-integers
Polar Coordinates - sin?

6. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Imaginary Numbers
Roots of Unity
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)

7. All numbers
transcendental
Field
complex
cos iy

8. In this amazing number field every algebraic equation in z with complex coefficients
conjugate pairs
-1
i^0
has a solution.

9. 2ib
Any polynomial O(xn) - (n > 0)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*

10. Rotates anticlockwise by p/2
interchangeable
Polar Coordinates - Multiplication by i
the distance from z to the origin in the complex plane
cos z

11. 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
imaginary
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
cosh²y - sinh²y

12. I
i^1
Field
Complex Conjugate
We say that c+di and c-di are complex conjugates.

13. R^2 = x
sin iy
Square Root
Any polynomial O(xn) - (n > 0)
cos z

14. A² + b² - real and non negative
cos iy
De Moivre's Theorem
sin z
zz*

15. The modulus of the complex number z= a + ib now can be interpreted as
Euler Formula
the distance from z to the origin 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)
irrational

16. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
non-integers
Rules of Complex Arithmetic
Complex Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)

17. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
e^(ln z)
Absolute Value of a Complex Number
radicals
the complex numbers

18. 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
the vector (a -b)
We say that c+di and c-di are complex conjugates.
Complex Multiplication
cos z

19. y / r
Polar Coordinates - sin?
(cos? +isin?)n
Polar Coordinates - Arg(z*)
Complex Exponentiation

20. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - sin?
Complex Exponentiation
non-integers
Field

21. Any number not rational
irrational
Complex Number
(a + c) + ( b + d)i
z - z*

22. All the powers of i can be written as
i²
four different numbers: i - -i - 1 - and -1.
Complex Number Formula
rational

23. I = imaginary unit - i² = -1 or i = v-1
zz*
z - z*
Polar Coordinates - z?¹
Imaginary Numbers

24. We see in this way that the distance between two points z and w in the complex plane is
rational
|z-w|
(a + c) + ( b + d)i
Polar Coordinates - Division

25. z1z2* / |z2|²
z1 / z2
Every complex number has the 'Standard Form': a + bi for some real a and b.
the distance from z to the origin in the complex plane
Euler Formula

26. Equivalent to an Imaginary Unit.
point of inflection
ln z
Polar Coordinates - Multiplication
Imaginary number

27. A plot of complex numbers as points.
Complex Numbers: Multiply
Complex Division
the distance from z to the origin in the complex plane
Argand diagram

28. Every complex number has the 'Standard Form':
a + bi for some real a and b.
z1 ^ (z2)
z1 / z2
How to add and subtract complex numbers (2-3i)-(4+6i)

29. 1
'i'
Complex Conjugate
Imaginary Numbers
i^2

30. For real a and b - a + bi =
i^4
interchangeable
Integers
0 if and only if a = b = 0

31. 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.'
Complex Subtraction
Euler's Formula
Rational Number
Complex Number

32. ? = -tan?
Liouville's Theorem -
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Division

33. I^2 =
conjugate pairs
-1
radicals
real

34. Like pi
Complex Number
transcendental
Argand diagram
standard form of complex numbers

35. The reals are just the
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane
Absolute Value of a Complex Number
the distance from z to the origin in the complex plane

36. 4th. Rule of Complex Arithmetic
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z

37. The square root of -1.
i^3
Polar Coordinates - sin?
Imaginary Unit
Subfield

38. x / r
Complex Numbers: Multiply
Polar Coordinates - cos?
i^2
z + z*

39. 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
We say that c+di and c-di are complex conjugates.
Complex Numbers: Add & subtract
Complex numbers are points in the plane
The Complex Numbers

40. A complex number may be taken to the power of another complex number.
standard form of complex numbers
Complex Exponentiation
a real number: (a + bi)(a - bi) = a² + b²
i^1

41. I
adding complex numbers
Liouville's Theorem -
v(-1)
i^2

42. A complex number and its conjugate
Polar Coordinates - r
adding complex numbers
ln z
conjugate pairs

43. A number that cannot be expressed as a fraction for any integer.
Euler's Formula
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Unit
Irrational Number

44. A subset within a field.
Subfield
z - z*
Complex Number
complex numbers

45. 1
a real number: (a + bi)(a - bi) = a² + b²
non-integers
cos z
i²

46. (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
Liouville's Theorem -
Subfield
Imaginary Numbers
How to solve (2i+3)/(9-i)

47. Multiply moduli and add arguments
transcendental
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Multiplication

48. The field of all rational and irrational numbers.
rational
The Complex Numbers
sin z
Real Numbers

49. Real and imaginary numbers
We say that c+di and c-di are complex conjugates.
multiply the numerator and the denominator by the complex conjugate of the denominator.
(cos? +isin?)n
complex numbers

50. Not on the numberline
subtracting complex numbers
transcendental
non-integers
adding complex numbers