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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. The field of all rational and irrational numbers.
The Complex Numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
imaginary
Real Numbers

2. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Polar Coordinates - z?¹
Square Root
(a + c) + ( b + d)i

3. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
rational
Imaginary Unit
Square Root

4. When two complex numbers are divided.
Complex Division
cos z
the distance from z to the origin in the complex plane
a + bi for some real a and b.

5. E ^ (z2 ln z1)
Polar Coordinates - r
z1 ^ (z2)
Absolute Value of a Complex Number
Integers

6. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Polar Coordinates - Division
Liouville's Theorem -
How to add and subtract complex numbers (2-3i)-(4+6i)

7. 3
Rational Number
interchangeable
How to add and subtract complex numbers (2-3i)-(4+6i)
i^3

8. A+bi
conjugate pairs
Argand diagram
complex numbers
Complex Number Formula

9. 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
i^2
complex numbers
cos z

10. A² + b² - real and non negative
zz*
adding complex numbers
Any polynomial O(xn) - (n > 0)
conjugate

11. Starts at 1 - does not include 0
natural
four different numbers: i - -i - 1 - and -1.
e^(ln z)
z + z*

12. No i
real
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula
Subfield

13. ½(e^(iz) + e^(-iz))
cos z
rational
Complex Subtraction
(cos? +isin?)n

14. Numbers on a numberline
Complex Addition
integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
v(-1)

15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
cos z
Polar Coordinates - z
Field
cos iy

16. 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
Rules of Complex Arithmetic
Liouville's Theorem -
Integers
standard form of complex numbers

17. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
v(-1)
multiplying complex numbers
Complex Numbers: Multiply
e^(ln z)

18. Equivalent to an Imaginary Unit.
Liouville's Theorem -
Integers
We say that c+di and c-di are complex conjugates.
Imaginary number

19. Multiply moduli and add arguments
Real and Imaginary Parts
|z-w|
Polar Coordinates - Multiplication
ln z

20. When two complex numbers are subtracted from one another.
Complex Subtraction
cosh²y - sinh²y
ln z
adding complex numbers

21. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Numbers: Multiply
Absolute Value of a Complex Number
real
Complex Exponentiation

22. V(zz*) = v(a² + b²)
conjugate pairs
|z| = mod(z)
point of inflection
can't get out of the complex numbers by adding (or subtracting) or multiplying two

23. ½(e^(-y) +e^(y)) = cosh y
z - z*
Complex Number Formula
z + z*
cos iy

24. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Complex Number Formula
i^2 = -1
z1 ^ (z2)

25. (e^(-y) - e^(y)) / 2i = i sinh y
a + bi for some real a and b.
sin iy
|z| = mod(z)
non-integers

26. Any number not rational
Square Root
Polar Coordinates - sin?
irrational
Complex numbers are points in the plane

27. (e^(iz) - e^(-iz)) / 2i
transcendental
Imaginary Numbers
sin z
Real and Imaginary Parts

28. For real a and b - a + bi =
can't get out of the complex numbers by adding (or subtracting) or multiplying two
0 if and only if a = b = 0
natural
a + bi for some real a and b.

29. 1
(a + c) + ( b + d)i
i²
Complex Multiplication
How to solve (2i+3)/(9-i)

30. The reals are just the
x-axis in the complex plane
the vector (a -b)
radicals
-1

31. Have radical
radicals
natural
point of inflection
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. A complex number may be taken to the power of another complex number.
Complex Exponentiation
i²
Complex Numbers: Add & subtract
zz*

33. 2ib
z - z*
Polar Coordinates - z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
-1

34. The complex number z representing a+bi.
(a + c) + ( b + d)i
De Moivre's Theorem
Polar Coordinates - Multiplication by i
Affix

35. R^2 = x
Complex numbers are points in the plane
i^2 = -1
Square Root
How to multiply complex nubers(2+i)(2i-3)

36. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
i^4
i²
sin z

37. A subset within a field.
We say that c+di and c-di are complex conjugates.
How to add and subtract complex numbers (2-3i)-(4+6i)
Subfield
transcendental

38. I
complex numbers
i^1
Complex numbers are points in the plane
Complex Addition

39. Rotates anticlockwise by p/2
the complex numbers
Imaginary Unit
natural
Polar Coordinates - Multiplication by i

40. (a + bi)(c + bi) =
For real a and b - a + bi = 0 if and only if a = b = 0
De Moivre's Theorem
sin iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

41. 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.'
has a solution.
sin iy
Any polynomial O(xn) - (n > 0)
Complex Number

42. 1
cosh²y - sinh²y
Imaginary Unit
Complex Exponentiation
Polar Coordinates - Multiplication

43. (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
How to multiply complex nubers(2+i)(2i-3)
cosh²y - sinh²y
'i'
natural

44. 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
a real number: (a + bi)(a - bi) = a² + b²
Complex Numbers: Add & subtract
z1 / z2
real

45. 1
i^2
irrational
Polar Coordinates - Division
Polar Coordinates - z

46. (a + bi) = (c + bi) =
(cos? +isin?)n
complex numbers
(a + c) + ( b + d)i
a + bi for some real a and b.

47. 2nd. Rule of Complex Arithmetic

48. We can also think of the point z= a+ ib as
Polar Coordinates - Division
Irrational Number
multiplying complex numbers
the vector (a -b)

49. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Complex Multiplication
a + bi for some real a and b.
Complex Division

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