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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. 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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2. Have radical
complex
i^1
radicals
Complex Division

3. 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.'
Argand diagram
interchangeable
Complex Number
Rules of Complex Arithmetic

4. The complex number z representing a+bi.
(cos? +isin?)n
Imaginary Numbers
rational
Affix

5. Cos n? + i sin n? (for all n integers)
multiplying complex numbers
the vector (a -b)
Polar Coordinates - Multiplication by i
(cos? +isin?)n

6. (a + bi) = (c + bi) =
Square Root
The Complex Numbers
(a + c) + ( b + d)i
Irrational Number

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

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8. (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
How to multiply complex nubers(2+i)(2i-3)
-1
radicals

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

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10. The modulus of the complex number z= a + ib now can be interpreted as
For real a and b - a + bi = 0 if and only if a = b = 0
i^0
The Complex Numbers
the distance from z to the origin in the complex plane

11. A complex number may be taken to the power of another complex number.
x-axis in the complex plane
four different numbers: i - -i - 1 - and -1.
point of inflection
Complex Exponentiation

12. 3rd. Rule of Complex Arithmetic
i^1
i²
the distance from z to the origin in the complex plane
For real a and b - a + bi = 0 if and only if a = b = 0

13. E ^ (z2 ln z1)
How to solve (2i+3)/(9-i)
z1 ^ (z2)
Complex Subtraction
Polar Coordinates - Division

14. ? = -tan?
Polar Coordinates - Arg(z*)
Complex Division
z1 ^ (z2)
imaginary

15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
sin iy
Integers
ln z
Complex Exponentiation

16. The field of all rational and irrational numbers.
standard form of complex numbers
Irrational Number
z + z*
Real Numbers

17. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Imaginary Numbers
e^(ln z)
Integers

18. Numbers on a numberline
Complex Subtraction
x-axis in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers

19. A+bi
|z| = mod(z)
cos iy
Complex Number Formula
i^0

20. All the powers of i can be written as
subtracting complex numbers
standard form of complex numbers
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.

21. The square root of -1.
How to multiply complex nubers(2+i)(2i-3)
the complex numbers
Imaginary Unit
Complex numbers are points in the plane

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
Complex Number
subtracting complex numbers
cosh²y - sinh²y
For real a and b - a + bi = 0 if and only if a = b = 0

23. Written as fractions - terminating + repeating decimals
Polar Coordinates - z?¹
rational
natural
i^1

24. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Rules of Complex Arithmetic
zz*
Euler's Formula

25. When two complex numbers are divided.
cos iy
Subfield
Complex Division
Imaginary Numbers

26. Root negative - has letter i
|z-w|
imaginary
For real a and b - a + bi = 0 if and only if a = b = 0
z1 / z2

27. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Argand diagram
sin iy
i²

28. Where the curvature of the graph changes
Real and Imaginary Parts
Imaginary Numbers
Polar Coordinates - Multiplication
point of inflection

29. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
real
standard form of complex numbers
(cos? +isin?)n

30. All numbers
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Division
complex

31. 1st. Rule of Complex Arithmetic
Polar Coordinates - z
i^2 = -1
Roots of Unity
Polar Coordinates - Multiplication

32. (e^(iz) - e^(-iz)) / 2i
z - z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
sin z
cos z

33. Like pi
ln z
Polar Coordinates - Multiplication by i
Liouville's Theorem -
transcendental

34. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Addition
Imaginary number
z - z*

35. Every complex number has the 'Standard Form':
-1
Complex Conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a + bi for some real a and b.

36. (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
Integers
How to solve (2i+3)/(9-i)
Real Numbers
i²

37. No i
real
conjugate pairs
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹

38. We can also think of the point z= a+ ib as
zz*
Complex Multiplication
Polar Coordinates - z
the vector (a -b)

39. A number that cannot be expressed as a fraction for any integer.
Irrational Number
cos iy
interchangeable
Complex Number Formula

40. x / r
Polar Coordinates - cos?
Every complex number has the 'Standard Form': a + bi for some real a and b.
natural
Complex Addition

41. 1
i²
-1
Polar Coordinates - r
sin z

42. (a + bi)(c + bi) =
i^3
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real and Imaginary Parts
Polar Coordinates - sin?

43. 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.
Complex Numbers: Multiply
Complex Subtraction
Real Numbers
the distance from z to the origin in the complex plane

44. A number that can be expressed as a fraction p/q where q is not equal to 0.
Affix
z + z*
Rational Number
Polar Coordinates - z

45. E^(ln r) e^(i?) e^(2pin)
(cos? +isin?)n
transcendental
e^(ln z)
sin iy

46. R^2 = x
How to solve (2i+3)/(9-i)
Square Root
Roots of Unity
sin iy

47. A + bi
Roots of Unity
the complex numbers
standard form of complex numbers
Polar Coordinates - z

48. The product of an imaginary number and its conjugate is
cos z
How to multiply complex nubers(2+i)(2i-3)
complex numbers
a real number: (a + bi)(a - bi) = a² + b²

49. 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
zz*
i^0
cos z
Real and Imaginary Parts

50. ½(e^(-y) +e^(y)) = cosh y
cos iy
Polar Coordinates - cos?
i^2 = -1
z1 / z2