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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. V(x² + y²) = |z|
Complex Numbers: Multiply
complex numbers
Polar Coordinates - r
z + z*

2. Numbers on a numberline
integers
Complex Numbers: Add & subtract
Polar Coordinates - z
How to add and subtract complex numbers (2-3i)-(4+6i)

3. 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.
cosh²y - sinh²y
cos iy
Complex numbers are points in the plane
i^1

4. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
radicals
Imaginary number
Complex Numbers: Multiply
Complex Subtraction

5. 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.
Polar Coordinates - cos?
How to find any Power
Polar Coordinates - Division
Subfield

6. V(zz*) = v(a² + b²)
Real Numbers
|z| = mod(z)
Polar Coordinates - Arg(z*)
(cos? +isin?)n

7. To simplify the square root of a negative number
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^3
natural

8. 2a
non-integers
z + z*
(a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0

9. R?¹(cos? - isin?)
sin iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
real

10. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Conjugate
Complex Multiplication
Liouville's Theorem -
How to add and subtract complex numbers (2-3i)-(4+6i)

11. I^2 =
-1
Polar Coordinates - r
complex
How to find any Power

12. The reals are just the
i^1
x-axis in the complex plane
-1
v(-1)

13. Written as fractions - terminating + repeating decimals
i^0
the distance from z to the origin in the complex plane
imaginary
rational

14. All numbers
rational
complex
x-axis in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.

15. When two complex numbers are added together.
i^3
Complex Addition
interchangeable
Polar Coordinates - cos?

16. 2ib
|z| = mod(z)
radicals
z - z*
has a solution.

17. All the powers of i can be written as
a real number: (a + bi)(a - bi) = a² + b²
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic
Complex Multiplication

18. Real and imaginary numbers
e^(ln z)
complex numbers
point of inflection
Field

19. Imaginary number

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20. 1
Imaginary Numbers
i^2
the vector (a -b)
Polar Coordinates - z?¹

21. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Complex Addition
natural
Liouville's Theorem -

22. 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
the vector (a -b)
'i'
Rules of Complex Arithmetic
Complex Multiplication

23. A number that cannot be expressed as a fraction for any integer.
Euler Formula
Irrational Number
Polar Coordinates - cos?
non-integers

24. E ^ (z2 ln z1)
Euler's Formula
Complex Division
Euler Formula
z1 ^ (z2)

25. (e^(-y) - e^(y)) / 2i = i sinh y
transcendental
sin iy
e^(ln z)
radicals

26. We can also think of the point z= a+ ib as
Argand diagram
the vector (a -b)
Polar Coordinates - sin?
i^0

27. (a + bi) = (c + bi) =
can't get out of the complex numbers by adding (or subtracting) or multiplying two
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
(a + c) + ( b + d)i

28. 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
v(-1)
Real and Imaginary Parts
Complex Number
Polar Coordinates - sin?

29. To simplify a complex fraction
Complex Conjugate
Imaginary Unit
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to add and subtract complex numbers (2-3i)-(4+6i)

30. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i^2 = -1
standard form of complex numbers
i^4
ln z

31. The field of all rational and irrational numbers.
Complex Number Formula
We say that c+di and c-di are complex conjugates.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real Numbers

32. 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
Polar Coordinates - z?¹
cos iy
the vector (a -b)

33. 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
Complex Numbers: Multiply
multiply the numerator and the denominator by the complex conjugate of the denominator.
the complex numbers
transcendental

34. Equivalent to an Imaginary Unit.
Complex Subtraction
Imaginary number
a + bi for some real a and b.
Rules of Complex Arithmetic

35. 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
cos iy
(cos? +isin?)n
Polar Coordinates - Division
adding complex numbers

36. 5th. Rule of Complex Arithmetic
|z-w|
rational
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

37. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r
Rules of Complex Arithmetic

38. Divide moduli and subtract arguments
Polar Coordinates - Division
Integers
the complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. When two complex numbers are divided.
Complex Division
rational
cos z
can't get out of the complex numbers by adding (or subtracting) or multiplying two

40. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
zz*
Integers
Polar Coordinates - z
Every complex number has the 'Standard Form': a + bi for some real a and b.

41. 1
Complex Conjugate
i^4
v(-1)
z1 ^ (z2)

42. A + bi
integers
Liouville's Theorem -
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers

43. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
We say that c+di and c-di are complex conjugates.
'i'
i^0

44. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Division
Imaginary number
non-integers
multiplying complex numbers

45. When two complex numbers are multipiled together.
Complex Multiplication
Complex Exponentiation
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex numbers

46. 2nd. Rule of Complex Arithmetic

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47. A² + b² - real and non negative
zz*
i^3
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

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

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49. Every complex number has the 'Standard Form':
How to solve (2i+3)/(9-i)
Rational Number
non-integers
a + bi for some real a and b.

50. (a + bi)(c + bi) =
complex
the complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.