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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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
cos z
Complex Number Formula
imaginary
standard form of complex numbers

2. E^(ln r) e^(i?) e^(2pin)
x-axis in the complex plane
Complex Conjugate
e^(ln z)
ln z

3. 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
zz*
Complex Number Formula
a + bi for some real a and b.

4. 2a
z1 ^ (z2)
z + z*
Real Numbers
four different numbers: i - -i - 1 - and -1.

5. Like pi
i^3
Polar Coordinates - cos?
complex numbers
transcendental

6. (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
subtracting complex numbers
v(-1)
Imaginary Unit
How to solve (2i+3)/(9-i)

7. Any number not rational
four different numbers: i - -i - 1 - and -1.
natural
irrational
Euler's Formula

8. A complex number may be taken to the power of another complex number.
ln z
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Exponentiation

9. (a + bi)(c + bi) =
Complex Conjugate
Imaginary number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic

10. 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
Complex Conjugate
0 if and only if a = b = 0
adding complex numbers
Polar Coordinates - Multiplication

11. Root negative - has letter i
imaginary
How to solve (2i+3)/(9-i)
subtracting complex numbers
Complex Number Formula

12. 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.'
For real a and b - a + bi = 0 if and only if a = b = 0
We say that c+di and c-di are complex conjugates.
integers
Complex Number

13. Derives z = a+bi
complex numbers
Complex Numbers: Multiply
Euler Formula
Imaginary Numbers

14. Given (4-2i) the complex conjugate would be (4+2i)
Imaginary number
Complex Conjugate
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0

15. Cos n? + i sin n? (for all n integers)
Affix
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n
Complex numbers are points in the plane

16. (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
Imaginary number
Complex Division
i^2
How to multiply complex nubers(2+i)(2i-3)

17. V(x² + y²) = |z|
Polar Coordinates - r
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply
De Moivre's Theorem

18. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
Argand diagram
irrational

19. 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
a real number: (a + bi)(a - bi) = a² + b²
We say that c+di and c-di are complex conjugates.
The Complex Numbers

20. When two complex numbers are multipiled together.
cosh²y - sinh²y
Complex Multiplication
Complex Exponentiation
Complex Subtraction

21. Real and imaginary numbers
complex numbers
cos z
How to find any Power
adding complex numbers

22. (a + bi) = (c + bi) =
Field
i^0
subtracting complex numbers
(a + c) + ( b + d)i

23. 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
conjugate pairs
complex
cos iy
Real and Imaginary Parts

24. Numbers on a numberline
|z| = mod(z)
Complex Multiplication
z1 ^ (z2)
integers

25. Imaginary number

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

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27. 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
Polar Coordinates - Multiplication by i
We say that c+di and c-di are complex conjugates.
real
Any polynomial O(xn) - (n > 0)

28. I
i^2 = -1
v(-1)
De Moivre's Theorem
real

29. 1
ln z
conjugate pairs
i^4
x-axis in the complex plane

30. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - Multiplication
Absolute Value of a Complex Number
multiplying complex numbers
sin z

31. 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
standard form of complex numbers
Real and Imaginary Parts
z + z*

32. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
i^3
Rules of Complex Arithmetic
Real and Imaginary Parts
Absolute Value of a Complex Number

33. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Arg(z*)
Euler's Formula
sin iy
interchangeable

34. A² + b² - real and non negative
complex
e^(ln z)
zz*
Every complex number has the 'Standard Form': a + bi for some real a and b.

35. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Rational Number
Euler Formula
Real Numbers

36. A + bi
real
standard form of complex numbers
integers
cos z

37. 2ib
the complex numbers
z - z*
i^2
z + z*

38. The complex number z representing a+bi.
|z-w|
Polar Coordinates - Division
zz*
Affix

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
natural
radicals
Complex Numbers: Add & subtract
(a + c) + ( b + d)i

40. In this amazing number field every algebraic equation in z with complex coefficients
real
has a solution.
transcendental
sin z

41. 3rd. Rule of Complex Arithmetic
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0
How to find any Power
i^2 = -1

42. 1st. Rule of Complex Arithmetic
Polar Coordinates - cos?
'i'
four different numbers: i - -i - 1 - and -1.
i^2 = -1

43. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
z1 / z2
adding complex numbers
Integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

44. 3
integers
i^3
|z| = mod(z)
Polar Coordinates - Arg(z*)

45. A number that cannot be expressed as a fraction for any integer.
Irrational Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.
|z| = mod(z)

46. ? = -tan?
Polar Coordinates - Arg(z*)
x-axis in the complex plane
Complex Numbers: Multiply
Complex Subtraction

47. The modulus of the complex number z= a + ib now can be interpreted as
How to add and subtract complex numbers (2-3i)-(4+6i)
Rational Number
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²

48. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - r
Polar Coordinates - cos?
The Complex Numbers
cos iy

49. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
Field
interchangeable

50. A complex number and its conjugate
Subfield
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2 = -1
conjugate pairs