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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 subset within a field.
a real number: (a + bi)(a - bi) = a² + b²
i^3
Subfield
We say that c+di and c-di are complex conjugates.

2. Cos n? + i sin n? (for all n integers)
Complex Division
(cos? +isin?)n
Polar Coordinates - sin?
Rational Number

3. To simplify a complex fraction
sin iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
Argand diagram
cosh²y - sinh²y

4. x / r
The Complex Numbers
Polar Coordinates - cos?
the vector (a -b)
Absolute Value of a Complex Number

5. E ^ (z2 ln z1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 ^ (z2)
Real Numbers
Rules of Complex Arithmetic

6. Have radical
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals
We say that c+di and c-di are complex conjugates.
natural

7. A number that cannot be expressed as a fraction for any integer.
conjugate
Irrational Number
non-integers
The Complex Numbers

8. When two complex numbers are multipiled together.
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
We say that c+di and c-di are complex conjugates.
'i'

9. (a + bi) = (c + bi) =
Complex Addition
Complex Number
Complex Numbers: Multiply
(a + c) + ( b + d)i

10. A complex number may be taken to the power of another complex number.
Complex Exponentiation
-1
For real a and b - a + bi = 0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)

11. y / r
Rules of Complex Arithmetic
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane
Polar Coordinates - sin?

12. We can also think of the point z= a+ ib as
i^1
Liouville's Theorem -
the vector (a -b)
Imaginary Numbers

13. (e^(iz) - e^(-iz)) / 2i
Absolute Value of a Complex Number
z + z*
sin z
Polar Coordinates - z?¹

14. A plot of complex numbers as points.
Any polynomial O(xn) - (n > 0)
Imaginary Numbers
Argand diagram
i^3

15. 2a
Euler's Formula
z + z*
(cos? +isin?)n
Roots of Unity

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

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17. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division
interchangeable
Roots of Unity

18. (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)
z + z*
cos z
How to multiply complex nubers(2+i)(2i-3)

19. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0

20. (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
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)
The Complex Numbers
irrational

21. The product of an imaginary number and its conjugate is
|z-w|
point of inflection
x-axis in the complex plane
a real number: (a + bi)(a - bi) = a² + b²

22. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - z?¹
irrational
z - z*
Rational Number

23. Every complex number has the 'Standard Form':
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
a real number: (a + bi)(a - bi) = a² + b²
non-integers

24. ? = -tan?
the distance from z to the origin in the complex plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Arg(z*)
Complex Number

25. When two complex numbers are divided.
Complex Division
Complex numbers are points in the plane
z1 / z2
|z| = mod(z)

26. Equivalent to an Imaginary Unit.
Polar Coordinates - sin?
Imaginary number
De Moivre's Theorem
Square Root

27. No i
Argand diagram
Real and Imaginary Parts
irrational
real

28. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - Multiplication by i
Irrational Number
four different numbers: i - -i - 1 - and -1.

29. 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
Complex Multiplication
Complex Number Formula

30. When two complex numbers are added together.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Addition
z + z*

31. Rotates anticlockwise by p/2
Imaginary Unit
interchangeable
e^(ln z)
Polar Coordinates - Multiplication by i

32. A² + b² - real and non negative
ln z
integers
zz*
e^(ln z)

33. Like pi
transcendental
Imaginary Numbers
irrational
Liouville's Theorem -

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

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35. All numbers
the complex numbers
complex
Irrational Number
interchangeable

36. Where the curvature of the graph changes
point of inflection
i²
Integers
cosh²y - sinh²y

37. Starts at 1 - does not include 0
natural
How to multiply complex nubers(2+i)(2i-3)
x-axis in the complex plane
sin z

38. The complex number z representing a+bi.
Affix
Complex Multiplication
Complex numbers are points in the plane
Square Root

39. Has exactly n roots by the fundamental theorem of algebra
four different numbers: i - -i - 1 - and -1.
has a solution.
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0

40. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
the distance from z to the origin in the complex plane
Absolute Value of a Complex Number
Polar Coordinates - Arg(z*)
standard form of complex numbers

41. When two complex numbers are subtracted from one another.
Complex Subtraction
Polar Coordinates - r
Complex Numbers: Multiply
Polar Coordinates - z?¹

42. The reals are just the
i^3
'i'
x-axis in the complex plane
z1 / z2

43. V(zz*) = v(a² + b²)
adding complex numbers
Roots of Unity
|z| = mod(z)
i^0

44. 1
cos z
i^0
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

45. Any number not rational
How to solve (2i+3)/(9-i)
Complex Numbers: Multiply
i^4
irrational

46. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
|z| = mod(z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Add & subtract
multiplying complex numbers

47. 2nd. Rule of Complex Arithmetic

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48. 1
Imaginary Unit
i^4
0 if and only if a = b = 0
(cos? +isin?)n

49. Written as fractions - terminating + repeating decimals
point of inflection
The Complex Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
rational

50. The field of all rational and irrational numbers.
irrational
rational
Real Numbers
z - z*