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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. R^2 = x
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication
Affix
Square Root

2. ½(e^(iz) + e^(-iz))
e^(ln z)
cos z
imaginary
Irrational Number

3. 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
i²
the complex numbers
Complex Number
a + bi for some real a and b.

4. When two complex numbers are divided.
i^4
Complex Division
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - r

5. The complex number z representing a+bi.
Subfield
Affix
the vector (a -b)
Polar Coordinates - Multiplication by i

6. To simplify a complex fraction
Every complex number has the 'Standard Form': a + bi for some real a and b.
conjugate pairs
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.

7. z1z2* / |z2|²
z + z*
Polar Coordinates - Arg(z*)
complex
z1 / z2

8. The modulus of the complex number z= a + ib now can be interpreted as
i^0
the distance from z to the origin in the complex plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit

9. Has exactly n roots by the fundamental theorem of algebra
i^2 = -1
ln z
Any polynomial O(xn) - (n > 0)
multiply the numerator and the denominator by the complex conjugate of the denominator.

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.'
Complex Number
Square Root
non-integers
has a solution.

11. 1
The Complex Numbers
cosh²y - sinh²y
adding complex numbers
Imaginary Unit

12. All numbers
complex
complex numbers
e^(ln z)
Irrational Number

13. 3
|z| = mod(z)
z1 ^ (z2)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^3

14. When two complex numbers are subtracted from one another.
Complex Division
Complex Subtraction
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - z?¹

15. 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.
real
'i'
Polar Coordinates - r
How to find any Power

16. A plot of complex numbers as points.
Complex numbers are points in the plane
interchangeable
Argand diagram
multiply the numerator and the denominator by the complex conjugate of the denominator.

17. 2nd. Rule of Complex Arithmetic

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18. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
conjugate pairs
Field
ln z
Complex Number

19. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply
z1 ^ (z2)
(cos? +isin?)n

20. 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
How to multiply complex nubers(2+i)(2i-3)
i^3
integers
Rules of Complex Arithmetic

21. Equivalent to an Imaginary Unit.
subtracting complex numbers
Rational Number
Irrational Number
Imaginary number

22. 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 add and subtract complex numbers (2-3i)-(4+6i)
Complex Numbers: Add & subtract
radicals
multiplying complex numbers

23. A subset within a field.
Polar Coordinates - r
Subfield
a + bi for some real a and b.
e^(ln z)

24. A number that can be expressed as a fraction p/q where q is not equal to 0.
real
z - z*
Rational Number
How to solve (2i+3)/(9-i)

25. A + bi
|z-w|
Complex Number
standard form of complex numbers
z - z*

26. The product of an imaginary number and its conjugate is
Roots of Unity
point of inflection
a + bi for some real a and b.
a real number: (a + bi)(a - bi) = a² + b²

27. Numbers on a numberline
imaginary
'i'
-1
integers

28. x + iy = r(cos? + isin?) = re^(i?)
Complex numbers are points in the plane
Polar Coordinates - z
z1 ^ (z2)
Polar Coordinates - Multiplication by i

29. E ^ (z2 ln z1)
z1 ^ (z2)
z + z*
cos z
Euler's Formula

30. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
the complex numbers
Integers
Euler Formula
z + z*

31. ? = -tan?
Polar Coordinates - z
the complex numbers
complex numbers
Polar Coordinates - Arg(z*)

32. No i
real
i^2
Polar Coordinates - sin?
How to find any Power

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
Polar Coordinates - cos?
subtracting complex numbers
i^2 = -1
Euler Formula

34. To simplify the square root of a negative number
x-axis in the complex plane
a + bi for some real a and b.
complex
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

35. In this amazing number field every algebraic equation in z with complex coefficients
multiplying complex numbers
Polar Coordinates - z?¹
has a solution.
Subfield

36. 1st. Rule of Complex Arithmetic
0 if and only if a = b = 0
Complex Addition
Any polynomial O(xn) - (n > 0)
i^2 = -1

37. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
sin z
We say that c+di and c-di are complex conjugates.
cos z
i^1

38. Every complex number has the 'Standard Form':
Imaginary number
a + bi for some real a and b.
ln z
'i'

39. 1
Square Root
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
non-integers

40. V(zz*) = v(a² + b²)
rational
|z| = mod(z)
cos z
Absolute Value of a Complex Number

41. All the powers of i can be written as
Argand diagram
four different numbers: i - -i - 1 - and -1.
sin z
Imaginary Unit

42. (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
How to solve (2i+3)/(9-i)
i^4
De Moivre's Theorem
Rational Number

43. (e^(iz) - e^(-iz)) / 2i
(a + c) + ( b + d)i
Imaginary Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z

44. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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45. 1
Euler's Formula
i^4
Polar Coordinates - r
Complex Multiplication

46. We see in this way that the distance between two points z and w in the complex plane is
four different numbers: i - -i - 1 - and -1.
Square Root
natural
|z-w|

47. (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
How to multiply complex nubers(2+i)(2i-3)
Complex numbers are points in the plane
Polar Coordinates - Multiplication
conjugate pairs

48. When two complex numbers are added together.
Complex Addition
i^3
Complex numbers are points in the plane
v(-1)

49. A+bi
Complex Number Formula
Real and Imaginary Parts
Imaginary number
Euler's Formula

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

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