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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. Any number not rational
Imaginary Numbers
subtracting complex numbers
irrational
0 if and only if a = b = 0

2. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Field
four different numbers: i - -i - 1 - and -1.
a real number: (a + bi)(a - bi) = a² + b²

3. E ^ (z2 ln z1)
We say that c+di and c-di are complex conjugates.
z1 ^ (z2)
(cos? +isin?)n
Polar Coordinates - z

4. When two complex numbers are added together.
Complex Addition
Any polynomial O(xn) - (n > 0)
Complex Conjugate
Polar Coordinates - Multiplication

5. 2ib
radicals
De Moivre's Theorem
z - z*
Polar Coordinates - cos?

6. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
natural
Complex Numbers: Add & subtract
Complex Addition

7. To simplify a complex fraction
four different numbers: i - -i - 1 - and -1.
z1 / z2
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real Numbers

8. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Field
non-integers
(a + c) + ( b + d)i

9. We see in this way that the distance between two points z and w in the complex plane is
ln z
|z-w|
i^1
sin z

10. 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.'
ln z
Complex Number
How to solve (2i+3)/(9-i)
complex numbers

11. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
the complex numbers
interchangeable
x-axis in the complex plane

12. A+bi
a real number: (a + bi)(a - bi) = a² + b²
natural
Polar Coordinates - sin?
Complex Number Formula

13. All numbers
z1 ^ (z2)
complex numbers
complex
point of inflection

14. For real a and b - a + bi =
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
cos z
Complex Numbers: Multiply

15. Cos n? + i sin n? (for all n integers)
has a solution.
natural
e^(ln z)
(cos? +isin?)n

16. Multiply moduli and add arguments
standard form of complex numbers
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication
Euler Formula

17. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²
irrational
sin iy

18. 2nd. Rule of Complex Arithmetic

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19. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply
integers
ln z

20. Divide moduli and subtract arguments
i^2 = -1
Euler's Formula
Field
Polar Coordinates - Division

21. 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
i^4
Real Numbers
i^2

22. (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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - cos?
Imaginary number
How to multiply complex nubers(2+i)(2i-3)

23. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
interchangeable
conjugate pairs
Euler's Formula

24. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^3
multiplying complex numbers
0 if and only if a = b = 0
Complex Conjugate

25. ½(e^(-y) +e^(y)) = cosh y
Complex Number
Polar Coordinates - r
cos iy
Polar Coordinates - sin?

26. y / r
the vector (a -b)
natural
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)

27. When two complex numbers are multipiled together.
Complex Number
Any polynomial O(xn) - (n > 0)
How to multiply complex nubers(2+i)(2i-3)
Complex Multiplication

28. The complex number z representing a+bi.
Real Numbers
Affix
the distance from z to the origin in the complex plane
Imaginary number

29. ½(e^(iz) + e^(-iz))
point of inflection
cos z
Subfield
Polar Coordinates - r

30. Root negative - has letter i
imaginary
Complex Numbers: Add & subtract
standard form of complex numbers
Imaginary Numbers

31. A number that can be expressed as a fraction p/q where q is not equal to 0.
sin z
Rational Number
Complex Number
imaginary

32. ? = -tan?
Integers
How to solve (2i+3)/(9-i)
Polar Coordinates - Arg(z*)
The Complex Numbers

33. 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
interchangeable
Real and Imaginary Parts
the complex numbers
Polar Coordinates - sin?

34. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rules of Complex Arithmetic
cos iy
z + z*

35. Like pi
Polar Coordinates - Multiplication
Polar Coordinates - sin?
transcendental
De Moivre's Theorem

36. Every complex number has the 'Standard Form':
a + bi for some real a and b.
De Moivre's Theorem
Polar Coordinates - Multiplication
Complex Multiplication

37. Rotates anticlockwise by p/2
Complex Subtraction
x-axis in the complex plane
conjugate
Polar Coordinates - Multiplication by i

38. The square root of -1.
Complex numbers are points in the plane
i²
Imaginary Unit
a real number: (a + bi)(a - bi) = a² + b²

39. The field of all rational and irrational numbers.
Rational Number
i^2
Real Numbers
irrational

40. A² + b² - real and non negative
zz*
Argand diagram
(a + c) + ( b + d)i
x-axis in the complex plane

41. 1
natural
i^2
Polar Coordinates - z
i²

42. (a + bi)(c + bi) =
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Exponentiation

43. V(zz*) = v(a² + b²)
|z| = mod(z)
ln z
transcendental
z1 / z2

44. (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
Any polynomial O(xn) - (n > 0)
standard form of complex numbers
How to solve (2i+3)/(9-i)
Imaginary Numbers

45. 1st. Rule of Complex Arithmetic
Polar Coordinates - Division
z + z*
i^2 = -1
cos iy

46. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Polar Coordinates - Arg(z*)
imaginary
Irrational Number

47. The modulus of the complex number z= a + ib now can be interpreted as
Subfield
the distance from z to the origin in the complex plane
Imaginary Numbers
Polar Coordinates - z?¹

48. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
ln z
Polar Coordinates - cos?
The Complex Numbers
Roots of Unity

49. When two complex numbers are subtracted from one another.
a real number: (a + bi)(a - bi) = a² + b²
Complex Subtraction
Rational Number
Roots of Unity

50. We can also think of the point z= a+ ib as
Polar Coordinates - z?¹
sin z
the vector (a -b)
integers