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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Multiplication
e^(ln z)
transcendental

2. E^(ln r) e^(i?) e^(2pin)
|z| = mod(z)
e^(ln z)
the complex numbers
'i'

3. For real a and b - a + bi =
real
Euler Formula
0 if and only if a = b = 0
a + bi for some real a and b.

4. To simplify a complex fraction
Complex Subtraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
interchangeable

5. Numbers on a numberline
real
integers
the vector (a -b)
Real Numbers

6. 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
i^0
Polar Coordinates - z
adding complex numbers
a real number: (a + bi)(a - bi) = a² + b²

7. The modulus of the complex number z= a + ib now can be interpreted as
Complex Addition
multiplying complex numbers
the distance from z to the origin in the complex plane
Absolute Value of a Complex Number

8. Equivalent to an Imaginary Unit.
standard form of complex numbers
Imaginary number
e^(ln z)
Polar Coordinates - Division

9. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
|z-w|
Roots of Unity
Field
Polar Coordinates - Multiplication

10. 4th. Rule of Complex Arithmetic
The Complex Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
De Moivre's Theorem
cos z

11. 1
Absolute Value of a Complex Number
Polar Coordinates - sin?
i²
conjugate pairs

12. 1
'i'
ln z
the vector (a -b)
i^4

13. V(x² + y²) = |z|
Polar Coordinates - r
Imaginary Numbers
Complex Numbers: Add & subtract
has a solution.

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

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15. All numbers
complex
The Complex Numbers
Imaginary Numbers
Imaginary Unit

16. Imaginary number

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17. ½(e^(-y) +e^(y)) = cosh y
Complex Subtraction
0 if and only if a = b = 0
zz*
cos iy

18. x + iy = r(cos? + isin?) = re^(i?)
i^0
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0
|z-w|

19. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
integers
Subfield
Complex Exponentiation

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
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?
How to multiply complex nubers(2+i)(2i-3)
Euler Formula

21. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Rules of Complex Arithmetic
ln z
sin z

22. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Imaginary number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
non-integers

23. 3rd. Rule of Complex Arithmetic
How to solve (2i+3)/(9-i)
e^(ln z)
transcendental
For real a and b - a + bi = 0 if and only if a = b = 0

24. Cos n? + i sin n? (for all n integers)
four different numbers: i - -i - 1 - and -1.
z1 / z2
cosh²y - sinh²y
(cos? +isin?)n

25. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Imaginary Numbers
multiplying complex numbers
natural
Every complex number has the 'Standard Form': a + bi for some real a and b.

26. Derives z = a+bi
v(-1)
real
Real and Imaginary Parts
Euler Formula

27. When two complex numbers are divided.
Complex Division
Argand diagram
integers
transcendental

28. Divide moduli and subtract arguments
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Division
has a solution.
Complex Number

29. I
i^1
Complex Division
i^0
De Moivre's Theorem

30. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Numbers: Add & subtract
conjugate pairs
the vector (a -b)
standard form of complex numbers

31. 2ib
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*
complex numbers

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

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33. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
Argand diagram
z - z*

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
Complex Numbers: Add & subtract
Rules of Complex Arithmetic
Complex Conjugate
the vector (a -b)

35. No i
Polar Coordinates - Multiplication by i
real
Square Root
four different numbers: i - -i - 1 - and -1.

36. R^2 = x
zz*
Imaginary number
Square Root
standard form of complex numbers

37. (e^(iz) - e^(-iz)) / 2i
sin z
Any polynomial O(xn) - (n > 0)
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

38. E ^ (z2 ln z1)
z1 ^ (z2)
Polar Coordinates - Multiplication by i
Real Numbers
Real and Imaginary Parts

39. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - cos?
Subfield
Complex Conjugate
Complex Number Formula

40. Rotates anticlockwise by p/2
Field
the vector (a -b)
i^0
Polar Coordinates - Multiplication by i

41. Starts at 1 - does not include 0
How to solve (2i+3)/(9-i)
real
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i

42. In this amazing number field every algebraic equation in z with complex coefficients
'i'
has a solution.
irrational
(cos? +isin?)n

43. The product of an imaginary number and its conjugate is
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²
Complex Exponentiation
Polar Coordinates - Multiplication by i

44. 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.
Field
i^4
Real Numbers
How to find any Power

45. When two complex numbers are multipiled together.
|z-w|
Complex Numbers: Add & subtract
Complex Multiplication
Liouville's Theorem -

46. A² + b² - real and non negative
zz*
i^3
the vector (a -b)
multiply the numerator and the denominator by the complex conjugate of the denominator.

47. A number that can be expressed as a fraction p/q where q is not equal to 0.
Imaginary number
multiplying complex numbers
Rational Number
integers

48. A number that cannot be expressed as a fraction for any integer.
Irrational Number
subtracting complex numbers
interchangeable
Polar Coordinates - Multiplication by i

49. Not on the numberline
the distance from z to the origin in the complex plane
non-integers
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication

50. V(zz*) = v(a² + b²)
For real a and b - a + bi = 0 if and only if a = b = 0
Integers
Complex Numbers: Multiply
|z| = mod(z)