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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. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
0 if and only if a = b = 0
Complex Numbers: Add & subtract
Imaginary Numbers

2. Imaginary number

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3. Have radical
Polar Coordinates - sin?
the distance from z to the origin in the complex plane
sin z
radicals

4. 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
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
non-integers
subtracting complex numbers

5. V(x² + y²) = |z|
Polar Coordinates - r
integers
cosh²y - sinh²y
Polar Coordinates - z

6. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
real
ln z
rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two

7. ? = -tan?
Complex Number
Polar Coordinates - Arg(z*)
How to find any Power
Any polynomial O(xn) - (n > 0)

8. (e^(iz) - e^(-iz)) / 2i
the distance from z to the origin in the complex plane
sin z
has a solution.
point of inflection

9. Starts at 1 - does not include 0
natural
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Euler Formula
z1 ^ (z2)

10. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
rational
Real Numbers
Absolute Value of a Complex Number
Complex Number

11. 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 Number
cos iy
the complex numbers
i^2

12. A + bi
standard form of complex numbers
four different numbers: i - -i - 1 - and -1.
Imaginary Numbers
Affix

13. Cos n? + i sin n? (for all n integers)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
complex numbers

14. 1
Field
i^0
the complex numbers
i^4

15. 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
Euler Formula
Rules of Complex Arithmetic
zz*

16. We see in this way that the distance between two points z and w in the complex plane is
0 if and only if a = b = 0
|z-w|
Field
multiply the numerator and the denominator by the complex conjugate of the denominator.

17. Where the curvature of the graph changes
Field
How to solve (2i+3)/(9-i)
Irrational Number
point of inflection

18. A complex number may be taken to the power of another complex number.
multiply the numerator and the denominator by the complex conjugate of the denominator.
imaginary
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Exponentiation

19. 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
-1
|z-w|
sin iy
Real and Imaginary Parts

20. We can also think of the point z= a+ ib as
How to solve (2i+3)/(9-i)
the vector (a -b)
conjugate
Polar Coordinates - Multiplication

21. All the powers of i can be written as
Field
How to find any Power
zz*
four different numbers: i - -i - 1 - and -1.

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

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23. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
the complex numbers
Field
The Complex Numbers
Polar Coordinates - Division

24. I
i^1
a real number: (a + bi)(a - bi) = a² + b²
Roots of Unity
standard form of complex numbers

25. Multiply moduli and add arguments
Argand diagram
Complex Multiplication
Polar Coordinates - Multiplication
The Complex Numbers

26. I
interchangeable
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
cos iy

27. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Complex Numbers: Add & subtract
i^2
point of inflection

28. A subset within a field.
natural
Subfield
e^(ln z)
Polar Coordinates - Multiplication by i

29. 3
|z| = mod(z)
We say that c+di and c-di are complex conjugates.
Roots of Unity
i^3

30. Written as fractions - terminating + repeating decimals
Polar Coordinates - Multiplication by i
Complex numbers are points in the plane
rational
i^4

31. (a + bi) = (c + bi) =
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)

32. A plot of complex numbers as points.
Polar Coordinates - sin?
|z| = mod(z)
Complex numbers are points in the plane
Argand diagram

33. V(zz*) = v(a² + b²)
|z| = mod(z)
transcendental
Real Numbers
We say that c+di and c-di are complex conjugates.

34. 3rd. Rule of Complex Arithmetic
(a + bi)(c + bi) = 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)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - r

35. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Field
Complex numbers are points in the plane
Any polynomial O(xn) - (n > 0)

36. 1
i^4
Liouville's Theorem -
multiplying complex numbers
non-integers

37. ½(e^(iz) + e^(-iz))
Square Root
cos z
'i'
Every complex number has the 'Standard Form': a + bi for some real a and b.

38. 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
transcendental
conjugate pairs
radicals

39. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Rational Number
Polar Coordinates - z
Complex Number

40. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
adding complex numbers
(cos? +isin?)n
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.

41. The field of all rational and irrational numbers.
Complex Addition
Real Numbers
real
x-axis in the complex plane

42. 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
cos z
Rules of Complex Arithmetic
z1 / z2
non-integers

43. x / r
Liouville's Theorem -
adding complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - cos?

44. 1
Imaginary number
e^(ln z)
Complex Multiplication
i²

45. A complex number and its conjugate
conjugate pairs
cosh²y - sinh²y
cos iy
standard form of complex numbers

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
Real and Imaginary Parts
How to solve (2i+3)/(9-i)
i^3
Polar Coordinates - cos?

47. No i
Integers
|z| = mod(z)
complex numbers
real

48. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
cos iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers

49. When two complex numbers are subtracted from one another.
a real number: (a + bi)(a - bi) = a² + b²
the vector (a -b)
Complex Subtraction
adding complex numbers

50. A² + b² - real and non negative
i^4
four different numbers: i - -i - 1 - and -1.
zz*
the distance from z to the origin in the complex plane