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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. (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
Affix
How to solve (2i+3)/(9-i)
Square Root
Rational Number

2. The modulus of the complex number z= a + ib now can be interpreted as
the vector (a -b)
standard form of complex numbers
i²
the distance from z to the origin in the complex plane

3. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
How to find any Power
multiplying complex numbers
Complex Conjugate
interchangeable

4. 1
Integers
conjugate
Complex Number
i^2

5. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
z - z*
The Complex Numbers
Complex Exponentiation

6. When two complex numbers are divided.
Complex Division
Imaginary Unit
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹

7. 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
z1 / z2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
adding complex numbers
i²

8. E^(ln r) e^(i?) e^(2pin)
point of inflection
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
Imaginary number

9. A² + b² - real and non negative
zz*
point of inflection
z + z*
has a solution.

10. Any number not rational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Absolute Value of a Complex Number
irrational
radicals

11. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Integers
the distance from z to the origin in the complex plane
Polar Coordinates - Division
Field

12. Like pi
transcendental
How to find any Power
i^0
rational

13. Imaginary number

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14. ? = -tan?
Polar Coordinates - Arg(z*)
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental

15. The reals are just the
Polar Coordinates - Division
x-axis in the complex plane
e^(ln z)
i^4

16. Cos n? + i sin n? (for all n integers)
Polar Coordinates - Division
(cos? +isin?)n
|z-w|
z1 / z2

17. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^2
Roots of Unity
'i'
-1

18. A complex number may be taken to the power of another complex number.
cosh²y - sinh²y
Real Numbers
Complex Exponentiation
i^2

19. To simplify the square root of a negative number
|z-w|
four different numbers: i - -i - 1 - and -1.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Arg(z*)

20. V(x² + y²) = |z|
i^4
Polar Coordinates - r
(cos? +isin?)n
Complex Addition

21. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
i^2
z - z*

22. Multiply moduli and add arguments
Complex Number Formula
(cos? +isin?)n
Liouville's Theorem -
Polar Coordinates - Multiplication

23. I^2 =
Integers
Polar Coordinates - r
-1
the distance from z to the origin in the complex plane

24. (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 Exponentiation
|z| = mod(z)
How to multiply complex nubers(2+i)(2i-3)
cos iy

25. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Polar Coordinates - Multiplication
Affix
multiply the numerator and the denominator by the complex conjugate of the denominator.

26. When two complex numbers are added together.
Complex Numbers: Add & subtract
Complex Addition
We say that c+di and c-di are complex conjugates.
sin z

27. The complex number z representing a+bi.
z - z*
sin z
natural
Affix

28. I
the complex numbers
How to solve (2i+3)/(9-i)
v(-1)
Irrational Number

29. ½(e^(iz) + e^(-iz))
Euler's Formula
cos z
Imaginary Unit
Polar Coordinates - Arg(z*)

30. 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
interchangeable
Polar Coordinates - Multiplication by i
Rules of Complex Arithmetic
The Complex Numbers

31. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary Numbers
z - z*

32. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to find any Power
ln z
sin iy
Rules of Complex Arithmetic

33. To simplify a complex fraction
e^(ln z)
Square Root
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

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

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35. 2nd. Rule of Complex Arithmetic

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36. 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
i^2
transcendental
Liouville's Theorem -

37. The field of all rational and irrational numbers.
Real Numbers
the vector (a -b)
Polar Coordinates - Arg(z*)
Rational Number

38. 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.
interchangeable
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z
Complex numbers are points in the plane

39. The square root of -1.
i^4
Imaginary Unit
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²

40. xpressions such as ``the complex number z'' - and ``the point z'' are now
natural
interchangeable
the distance from z to the origin in the complex plane
i^2 = -1

41. 3
complex
non-integers
i^3
Square Root

42. 1st. Rule of Complex Arithmetic
has a solution.
How to find any Power
Complex Numbers: Multiply
i^2 = -1

43. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Arg(z*)
Complex Numbers: Multiply
a real number: (a + bi)(a - bi) = a² + b²
The Complex Numbers

44. 5th. Rule of Complex Arithmetic
rational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler's Formula
Square Root

45. All numbers
has a solution.
Complex Exponentiation
interchangeable
complex

46. V(zz*) = v(a² + b²)
the vector (a -b)
|z| = mod(z)
i²
Irrational Number

47. Divide moduli and subtract arguments
|z-w|
Square Root
v(-1)
Polar Coordinates - Division

48. E ^ (z2 ln z1)
the vector (a -b)
Complex Number
z1 ^ (z2)
Complex Exponentiation

49. x / r
(a + c) + ( b + d)i
-1
Polar Coordinates - cos?
conjugate pairs

50. When two complex numbers are subtracted from one another.
Complex Subtraction
How to find any Power
Polar Coordinates - Division
Polar Coordinates - cos?