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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. Multiply moduli and add arguments
zz*
transcendental
Polar Coordinates - Multiplication
Imaginary number

2. 1
Polar Coordinates - Multiplication
(cos? +isin?)n
i²
i^0

3. 4th. Rule of Complex Arithmetic
sin iy
multiplying complex numbers
Complex numbers are points in the plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i

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

5. 2ib
point of inflection
0 if and only if a = b = 0
z - z*
Irrational Number

6. The product of an imaginary number and its conjugate is
Complex Number
a real number: (a + bi)(a - bi) = a² + b²
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - sin?

7. Real and imaginary numbers
z + z*
complex numbers
Any polynomial O(xn) - (n > 0)
Complex Multiplication

8. The field of all rational and irrational numbers.
i^3
Real Numbers
Polar Coordinates - Arg(z*)
Euler Formula

9. (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
i²
How to multiply complex nubers(2+i)(2i-3)
Square Root

10. ½(e^(iz) + e^(-iz))
cos z
z1 / z2
a real number: (a + bi)(a - bi) = a² + b²
i²

11. Numbers on a numberline
De Moivre's Theorem
integers
Polar Coordinates - z?¹
Real and Imaginary Parts

12. Where the curvature of the graph changes
a + bi for some real a and b.
point of inflection
Field
four different numbers: i - -i - 1 - and -1.

13. A plot of complex numbers as points.
Affix
Argand diagram
i^0
Euler's Formula

14. z1z2* / |z2|²
Imaginary number
z1 / z2
x-axis in the complex plane
complex

15. I^2 =
-1
subtracting complex numbers
Polar Coordinates - cos?
Polar Coordinates - Arg(z*)

16. A + bi
integers
standard form of complex numbers
e^(ln z)
Complex Number Formula

17. A number that can be expressed as a fraction p/q where q is not equal to 0.
has a solution.
rational
Rational Number
Euler Formula

18. x + iy = r(cos? + isin?) = re^(i?)
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0
De Moivre's Theorem
Polar Coordinates - z

19. Imaginary number

20. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Multiplication
complex numbers
Integers
Complex Exponentiation

21. To simplify a complex fraction
zz*
Polar Coordinates - sin?
z1 / z2
multiply the numerator and the denominator by the complex conjugate of the denominator.

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

23. Derives z = a+bi
Roots of Unity
Euler Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiply the numerator and the denominator by the complex conjugate of the denominator.

24. When two complex numbers are added together.
interchangeable
Complex Addition
sin iy
Liouville's Theorem -

25. (a + bi)(c + bi) =
Polar Coordinates - r
complex numbers
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. 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
|z| = mod(z)
Irrational Number
the complex numbers
Imaginary Numbers

27. Every complex number has the 'Standard Form':
a + bi for some real a and b.
irrational
subtracting complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

28. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
x-axis in the complex plane
Polar Coordinates - Multiplication by i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

29. Have radical
sin z
interchangeable
radicals
standard form of complex numbers

30. x / r
i^4
Polar Coordinates - cos?
Real and Imaginary Parts
non-integers

31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
i^4
The Complex Numbers
non-integers
(a + c) + ( b + d)i

32. ? = -tan?
Integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Arg(z*)

33. 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
subtracting complex numbers
complex
Polar Coordinates - sin?
Rules of Complex Arithmetic

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
Polar Coordinates - Arg(z*)
Rules of Complex Arithmetic
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i

35. A complex number may be taken to the power of another complex number.
i²
0 if and only if a = b = 0
We say that c+di and c-di are complex conjugates.
Complex Exponentiation

36. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
multiply the numerator and the denominator by the complex conjugate of the denominator.

37. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
non-integers
|z| = mod(z)
conjugate
Any polynomial O(xn) - (n > 0)

38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Imaginary Unit
Roots of Unity
Euler's Formula
Affix

39. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
e^(ln z)
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

40. A² + b² - real and non negative
Complex Number Formula
zz*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate

41. All numbers
subtracting complex numbers
(cos? +isin?)n
complex
(a + bi) = (c + bi) = (a + c) + ( b + d)i

42. Equivalent to an Imaginary Unit.
cos z
a + bi for some real a and b.
|z-w|
Imaginary number

43. 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
(cos? +isin?)n
Real and Imaginary Parts
Argand diagram
|z-w|

44. Any number not rational
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers
the vector (a -b)
irrational

45. We can also think of the point z= a+ ib as
the vector (a -b)
interchangeable
complex
a + bi for some real a and b.

46. (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
Imaginary number
How to solve (2i+3)/(9-i)
standard form of complex numbers
z - z*

47. E^(ln r) e^(i?) e^(2pin)
We say that c+di and c-di are complex conjugates.
e^(ln z)
a real number: (a + bi)(a - bi) = a² + b²
a + bi for some real a and b.

48. 1
Affix
Absolute Value of a Complex Number
i^4
z1 / z2

49. I
Liouville's Theorem -
Square Root
i^1
integers

50. The reals are just the
standard form of complex numbers
x-axis in the complex plane
imaginary
Polar Coordinates - Multiplication