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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
ln z
cos iy
Square Root

2. 4th. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
How to multiply complex nubers(2+i)(2i-3)

3. Root negative - has letter i
Liouville's Theorem -
imaginary
i^2
z1 ^ (z2)

4. 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
irrational
The Complex Numbers
We say that c+di and c-di are complex conjugates.
-1

5. R?¹(cos? - isin?)
irrational
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z?¹
Polar Coordinates - Division

6. 1
i²
integers
four different numbers: i - -i - 1 - and -1.
z + z*

7. E ^ (z2 ln z1)
Complex Division
cosh²y - sinh²y
Imaginary Numbers
z1 ^ (z2)

8. Equivalent to an Imaginary Unit.
How to multiply complex nubers(2+i)(2i-3)
Complex Subtraction
Imaginary number
cos iy

9. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
i^1
-1
i²
Complex Numbers: Multiply

10. To simplify the square root of a negative number
0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i

11. The square root of -1.
z1 ^ (z2)
Imaginary Unit
four different numbers: i - -i - 1 - and -1.
x-axis in the complex plane

12. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
conjugate
Liouville's Theorem -
complex
Field

13. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Arg(z*)
De Moivre's Theorem
How to solve (2i+3)/(9-i)
e^(ln z)

14. ? = -tan?
Polar Coordinates - Arg(z*)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler's Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two

15. Divide moduli and subtract arguments
Polar Coordinates - Division
zz*
Euler Formula
Integers

16. Have radical
transcendental
Complex Multiplication
radicals
(a + c) + ( b + d)i

17. 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
Subfield
Real and Imaginary Parts
Liouville's Theorem -
Euler Formula

18. Written as fractions - terminating + repeating decimals
Integers
-1
Rules of Complex Arithmetic
rational

19. x + iy = r(cos? + isin?) = re^(i?)
Roots of Unity
Argand diagram
Polar Coordinates - z
adding complex numbers

20. z1z2* / |z2|²
z + z*
imaginary
z1 / z2
i²

21. (e^(iz) - e^(-iz)) / 2i
Square Root
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin z
natural

22. 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
zz*
Irrational Number
Polar Coordinates - z

23. Where the curvature of the graph changes
a real number: (a + bi)(a - bi) = a² + b²
point of inflection
z + z*
i^4

24. Every complex number has the 'Standard Form':
a + bi for some real a and b.
ln z
We say that c+di and c-di are complex conjugates.
How to add and subtract complex numbers (2-3i)-(4+6i)

25. (a + bi) = (c + bi) =
natural
The Complex Numbers
(a + c) + ( b + d)i
conjugate pairs

26. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Complex Addition
How to multiply complex nubers(2+i)(2i-3)
i^4

27. The field of all rational and irrational numbers.
a real number: (a + bi)(a - bi) = a² + b²
i^1
Real Numbers
z - z*

28. 3rd. Rule of Complex Arithmetic
point of inflection
i^2
For real a and b - a + bi = 0 if and only if a = b = 0
cos iy

29. V(x² + y²) = |z|
We say that c+di and c-di are complex conjugates.
Polar Coordinates - r
interchangeable
imaginary

30. Imaginary number

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31. 2ib
Complex Addition
z - z*
Imaginary number
0 if and only if a = b = 0

32. 3
i^3
i^1
Euler Formula
i^4

33. A complex number may be taken to the power of another complex number.
complex numbers
non-integers
Complex Exponentiation
i^3

34. 1st. Rule of Complex Arithmetic
(a + c) + ( b + d)i
i^2 = -1
z1 ^ (z2)
non-integers

35. A number that cannot be expressed as a fraction for any integer.
Complex Subtraction
sin z
ln z
Irrational Number

36. ½(e^(-y) +e^(y)) = cosh y
Complex Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real and Imaginary Parts
cos iy

37. 1
Absolute Value of a Complex Number
Euler Formula
i^4
Roots of Unity

38. When two complex numbers are added together.
z1 / z2
the vector (a -b)
Complex Addition
Polar Coordinates - z

39. Not on the numberline
a + bi for some real a and b.
Field
a real number: (a + bi)(a - bi) = a² + b²
non-integers

40. The product of an imaginary number and its conjugate is
Any polynomial O(xn) - (n > 0)
-1
a real number: (a + bi)(a - bi) = a² + b²
z1 ^ (z2)

41. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
the distance from z to the origin in the complex plane

42. When two complex numbers are divided.
a + bi for some real a and b.
Complex Division
For real a and b - a + bi = 0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²

43. 5th. Rule of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
interchangeable
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

44. (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)
Polar Coordinates - cos?
Liouville's Theorem -
transcendental

45. 2a
i^2
sin iy
z + z*
has a solution.

46. A + bi
For real a and b - a + bi = 0 if and only if a = b = 0
standard form of complex numbers
v(-1)
Polar Coordinates - r

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

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48. Any number not rational
irrational
Rational Number
The Complex Numbers
sin iy

49. 2nd. Rule of Complex Arithmetic

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50. Derives z = a+bi
Euler Formula
Complex Numbers: Multiply
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - r