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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. Like pi
transcendental
a real number: (a + bi)(a - bi) = a² + b²
Any polynomial O(xn) - (n > 0)
Polar Coordinates - sin?

2. When two complex numbers are divided.
How to solve (2i+3)/(9-i)
conjugate pairs
Complex Division
Any polynomial O(xn) - (n > 0)

3. (a + bi) = (c + bi) =
Real and Imaginary Parts
(a + c) + ( b + d)i
Subfield
0 if and only if a = b = 0

4. 1st. Rule of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers
x-axis in the complex plane
i^2 = -1

5. 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
Polar Coordinates - Multiplication by i
Complex Number Formula
Complex Numbers: Multiply

6. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Complex Multiplication
Field
subtracting complex numbers

7. (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)
Complex Exponentiation
multiplying complex numbers
non-integers

8. The reals are just the
has a solution.
Argand diagram
Polar Coordinates - Arg(z*)
x-axis in the complex plane

9. To simplify the square root of a negative number
Subfield
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
De Moivre's Theorem

10. 2a
cos iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiplying complex numbers
z + z*

11. All the powers of i can be written as
Complex Exponentiation
four different numbers: i - -i - 1 - and -1.
Imaginary Numbers
conjugate pairs

12. Numbers on a numberline
Polar Coordinates - Division
i^2 = -1
ln z
integers

13. 1
i^4
Real and Imaginary Parts
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Absolute Value of a Complex Number

14. The field of all rational and irrational numbers.
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.
x-axis in the complex plane
Real Numbers

15. 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
How to add and subtract complex numbers (2-3i)-(4+6i)
interchangeable
We say that c+di and c-di are complex conjugates.
Polar Coordinates - sin?

16. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
|z| = mod(z)
i^1
How to solve (2i+3)/(9-i)

17. 1
natural
the vector (a -b)
i²
Imaginary number

18. ½(e^(-y) +e^(y)) = cosh y
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number
cos iy
Rules of Complex Arithmetic

19. 2ib
z - z*
transcendental
rational
the complex numbers

20. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - sin?
the complex numbers
Complex Conjugate
Polar Coordinates - z?¹

21. A complex number and its conjugate
conjugate pairs
cosh²y - sinh²y
irrational
z - z*

22. R^2 = x
Affix
Complex Numbers: Add & subtract
(cos? +isin?)n
Square Root

23. Starts at 1 - does not include 0
natural
cos iy
transcendental
(a + c) + ( b + d)i

24. When two complex numbers are subtracted from one another.
Complex Subtraction
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
complex numbers
The Complex Numbers

25. (e^(iz) - e^(-iz)) / 2i
multiplying complex numbers
-1
Field
sin z

26. Not on the numberline
multiply the numerator and the denominator by the complex conjugate of the denominator.
non-integers
How to find any Power
the vector (a -b)

27. Derives z = a+bi
Complex numbers are points in the plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary Numbers
Euler Formula

28. 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
i^3
Rational Number
'i'
Rules of Complex Arithmetic

29. In this amazing number field every algebraic equation in z with complex coefficients
i^0
Affix
has a solution.
i^2

30. All numbers
ln z
complex
sin iy
Polar Coordinates - Multiplication

31. 1
standard form of complex numbers
i^4
i^2
cosh²y - sinh²y

32. x / r
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0
Irrational Number
x-axis in the complex plane

33. A complex number may be taken to the power of another complex number.
transcendental
Complex Exponentiation
Imaginary Unit
z1 / z2

34. 5th. Rule of Complex Arithmetic
the complex numbers
Imaginary number
Argand diagram
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

35. Written as fractions - terminating + repeating decimals
rational
Imaginary number
Complex Subtraction
irrational

36. A number that cannot be expressed as a fraction for any integer.
0 if and only if a = b = 0
Rational Number
Irrational Number
the vector (a -b)

37. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
i^4
Subfield
i^1

38. ? = -tan?
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Arg(z*)
Imaginary Numbers
irrational

39. 1
i^1
i^0
Liouville's Theorem -
cosh²y - sinh²y

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

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41. Equivalent to an Imaginary Unit.
cosh²y - sinh²y
e^(ln z)
conjugate pairs
Imaginary number

42. The product of an imaginary number and its conjugate is
the vector (a -b)
Square Root
(cos? +isin?)n
a real number: (a + bi)(a - bi) = a² + b²

43. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Complex Numbers: Multiply
Polar Coordinates - z
Roots of Unity
For real a and b - a + bi = 0 if and only if a = b = 0

44. 2nd. Rule of Complex Arithmetic

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45. Any number not rational
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
irrational
Absolute Value of a Complex Number

46. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Irrational Number
has a solution.
How to add and subtract complex numbers (2-3i)-(4+6i)
radicals

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

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48. E ^ (z2 ln z1)
Real and Imaginary Parts
real
z1 ^ (z2)
i^1

49. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
e^(ln z)
Complex Division
Affix

50. V(zz*) = v(a² + b²)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z| = mod(z)
complex
How to multiply complex nubers(2+i)(2i-3)