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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. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Polar Coordinates - z?¹
cos iy
(a + c) + ( b + d)i
conjugate

2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
cos z
Rational Number
For real a and b - a + bi = 0 if and only if a = b = 0

3. Given (4-2i) the complex conjugate would be (4+2i)
How to solve (2i+3)/(9-i)
Argand diagram
Complex Conjugate
cos iy

4. R?¹(cos? - isin?)
Polar Coordinates - Multiplication by i
Complex Number Formula
Polar Coordinates - z?¹
Polar Coordinates - Arg(z*)

5. Numbers on a numberline
integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
We say that c+di and c-di are complex conjugates.
Rules of Complex Arithmetic

6. E^(ln r) e^(i?) e^(2pin)
0 if and only if a = b = 0
e^(ln z)
conjugate
Complex Addition

7. 2nd. Rule of Complex Arithmetic

8. 1st. Rule of Complex Arithmetic
i^2 = -1
Subfield
standard form of complex numbers
has a solution.

9. A² + b² - real and non negative
Roots of Unity
zz*
a + bi for some real a and b.
Liouville's Theorem -

10. Every complex number has the 'Standard Form':
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiplying complex numbers
sin iy
a + bi for some real a and b.

11. When two complex numbers are multipiled together.
x-axis in the complex plane
How to multiply complex nubers(2+i)(2i-3)
Complex Multiplication
Polar Coordinates - Multiplication by i

12. (a + bi)(c + bi) =
Real and Imaginary Parts
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
Square Root

13. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Argand diagram
Polar Coordinates - z?¹
the complex numbers
Absolute Value of a Complex Number

14. Derives z = a+bi
Complex Number Formula
z + z*
Euler Formula
Absolute Value of a Complex Number

15. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Arg(z*)
sin z
0 if and only if a = b = 0
e^(ln z)

16. To simplify a complex fraction
Polar Coordinates - r
Euler's Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power

17. R^2 = x
Square Root
i^0
Polar Coordinates - sin?
x-axis in the complex plane

18. 2a
(cos? +isin?)n
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cos z

19. 2ib
z - z*
rational
irrational
non-integers

20. ? = -tan?
Polar Coordinates - Arg(z*)
Complex Multiplication
Polar Coordinates - z?¹
Every complex number has the 'Standard Form': a + bi for some real a and b.

21. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
transcendental
Complex Exponentiation
can't get out of the complex numbers by adding (or subtracting) or multiplying two

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

23. We can also think of the point z= a+ ib as
Polar Coordinates - Arg(z*)
e^(ln z)
the vector (a -b)
has a solution.

24. Divide moduli and subtract arguments
Complex Subtraction
Polar Coordinates - Division
complex
Euler Formula

25. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - Multiplication by i
i^0
How to add and subtract complex numbers (2-3i)-(4+6i)
v(-1)

26. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Euler's Formula
imaginary
Polar Coordinates - z?¹

27. The reals are just the
Every complex number has the 'Standard Form': a + bi for some real a and b.
x-axis in the complex plane
four different numbers: i - -i - 1 - and -1.
zz*

28. The field of all rational and irrational numbers.
Rational Number
Real Numbers
Polar Coordinates - Multiplication
transcendental

29. Written as fractions - terminating + repeating decimals
has a solution.
Complex Numbers: Add & subtract
integers
rational

30. When two complex numbers are subtracted from one another.
Complex Subtraction
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2
the vector (a -b)

31. In this amazing number field every algebraic equation in z with complex coefficients
z1 / z2
Polar Coordinates - sin?
irrational
has a solution.

32. A subset within a field.
Field
Subfield
Polar Coordinates - Arg(z*)
adding complex numbers

33. Equivalent to an Imaginary Unit.
complex
Imaginary number
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0

34. Real and imaginary numbers
complex numbers
How to solve (2i+3)/(9-i)
Complex Subtraction
cosh²y - sinh²y

35. I = imaginary unit - i² = -1 or i = v-1
How to multiply complex nubers(2+i)(2i-3)
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers

36. 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
conjugate pairs
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers

37. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - r
Affix

38. 1
sin iy
cosh²y - sinh²y
Rules of Complex Arithmetic
cos z

39. ½(e^(iz) + e^(-iz))
cos z
conjugate
Subfield
imaginary

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

41. A complex number and its conjugate
Complex Numbers: Add & subtract
conjugate pairs
cosh²y - sinh²y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

42. To simplify the square root of a negative number
Polar Coordinates - z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.
z1 / z2

43. All numbers
complex
a real number: (a + bi)(a - bi) = a² + b²
Complex Division
sin z

44. 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
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real and Imaginary Parts

45. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

46. 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
Irrational Number
subtracting complex numbers
i^1

47. z1z2* / |z2|²
Complex Multiplication
z1 / z2
Euler's Formula
Liouville's Theorem -

48. The modulus of the complex number z= a + ib now can be interpreted as
standard form of complex numbers
Complex Numbers: Add & subtract
Complex Multiplication
the distance from z to the origin in the complex plane

49. xpressions such as ``the complex number z'' - and ``the point z'' are now
z - z*
interchangeable
x-axis in the complex plane
i^2

50. When two complex numbers are added together.
The Complex Numbers
Complex Addition
i^2 = -1
Absolute Value of a Complex Number