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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. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
cos z
Real and Imaginary Parts
i^2 = -1

2. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
How to solve (2i+3)/(9-i)
the complex numbers

3. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
z - z*
cosh²y - sinh²y
How to find any Power
Absolute Value of a Complex Number

4. x + iy = r(cos? + isin?) = re^(i?)
i^2 = -1
|z| = mod(z)
cos iy
Polar Coordinates - z

5. Divide moduli and subtract arguments
Polar Coordinates - Division
zz*
Imaginary Unit
Subfield

6. Any number not rational
(cos? +isin?)n
Complex Numbers: Multiply
irrational
Polar Coordinates - Multiplication by i

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

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8. 1
Complex Numbers: Multiply
i²
Absolute Value of a Complex Number
z1 / z2

9. 5th. Rule of Complex Arithmetic
Complex Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection
Complex Exponentiation

10. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^1
interchangeable
Polar Coordinates - Division
Polar Coordinates - sin?

11. A² + b² - real and non negative
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*
Complex Numbers: Multiply
Polar Coordinates - Arg(z*)

12. 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.
conjugate pairs
cos iy
e^(ln z)
Complex numbers are points in the plane

13. A+bi
Affix
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula
Complex Numbers: Multiply

14. A subset within a field.
Subfield
radicals
multiplying complex numbers
How to solve (2i+3)/(9-i)

15. The product of an imaginary number and its conjugate is
v(-1)
i^2 = -1
a real number: (a + bi)(a - bi) = a² + b²
Complex Number

16. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z
standard form of complex numbers

17. ? = -tan?
Polar Coordinates - Arg(z*)
has a solution.
We say that c+di and c-di are complex conjugates.
Argand diagram

18. Written as fractions - terminating + repeating decimals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
rational
Complex Numbers: Add & subtract

19. When two complex numbers are multipiled together.
Any polynomial O(xn) - (n > 0)
imaginary
cos iy
Complex Multiplication

20. (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
0 if and only if a = b = 0
adding complex numbers
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane

21. V(zz*) = v(a² + b²)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
|z| = mod(z)
-1

22. When two complex numbers are added together.
subtracting complex numbers
Complex Addition
Roots of Unity
Imaginary Numbers

23. Rotates anticlockwise by p/2
cosh²y - sinh²y
e^(ln z)
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Multiplication by i

24. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
conjugate
Complex Multiplication
How to find any Power
complex

25. Imaginary number

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26. 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.
integers
Complex Numbers: Multiply
z - z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. I
v(-1)
radicals
Polar Coordinates - z?¹
z1 / z2

28. E^(ln r) e^(i?) e^(2pin)
Field
v(-1)
e^(ln z)
(a + c) + ( b + d)i

29. A number that can be expressed as a fraction p/q where q is not equal to 0.
(cos? +isin?)n
Polar Coordinates - z
Field
Rational Number

30. A + bi
standard form of complex numbers
Integers
subtracting complex numbers
Real Numbers

31. 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
Complex Addition
i^0
De Moivre's Theorem

32. 1st. Rule of Complex Arithmetic
Complex numbers are points in the plane
i^0
the distance from z to the origin in the complex plane
i^2 = -1

33. We can also think of the point z= a+ ib as
the vector (a -b)
standard form of complex numbers
conjugate
z + z*

34. (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
i^2 = -1
How to multiply complex nubers(2+i)(2i-3)
Square Root
Complex numbers are points in the plane

35. Given (4-2i) the complex conjugate would be (4+2i)
natural
imaginary
0 if and only if a = b = 0
Complex Conjugate

36. To simplify a complex fraction
Every complex number has the 'Standard Form': a + bi for some real a and b.
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^0
Real and Imaginary Parts

37. A complex number and its conjugate
multiplying complex numbers
Real and Imaginary Parts
conjugate pairs
(cos? +isin?)n

38. 1
Square Root
complex numbers
cosh²y - sinh²y
Polar Coordinates - Arg(z*)

39. 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
Complex Addition
Polar Coordinates - Arg(z*)
Real and Imaginary Parts
We say that c+di and c-di are complex conjugates.

40. I^2 =
-1
zz*
Complex Number
Complex Multiplication

41. Multiply moduli and add arguments
(cos? +isin?)n
Polar Coordinates - Multiplication
Rules of Complex Arithmetic
Polar Coordinates - sin?

42. A plot of complex numbers as points.
Argand diagram
Complex Numbers: Multiply
sin z
Polar Coordinates - z?¹

43. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
cos iy
Imaginary Unit
Liouville's Theorem -

44. All the powers of i can be written as
i^1
Real and Imaginary Parts
Complex Number
four different numbers: i - -i - 1 - and -1.

45. The field of all rational and irrational numbers.
interchangeable
|z| = mod(z)
z + z*
Real Numbers

46. All numbers
complex
Polar Coordinates - Multiplication by i
Rules of Complex Arithmetic
v(-1)

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

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48. 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
Real and Imaginary Parts
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
radicals

49. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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50. A complex number may be taken to the power of another complex number.
cosh²y - sinh²y
We say that c+di and c-di are complex conjugates.
radicals
Complex Exponentiation