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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. Imaginary number

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2. Rotates anticlockwise by p/2
Complex Subtraction
Euler's Formula
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply

3. Derives z = a+bi
Roots of Unity
Euler Formula
Integers
We say that c+di and c-di are complex conjugates.

4. 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
Complex Subtraction
the complex numbers
We say that c+di and c-di are complex conjugates.
i^0

5. A complex number may be taken to the power of another complex number.
Complex Number
Complex Exponentiation
i^2
Complex Conjugate

6. V(zz*) = v(a² + b²)
Affix
i^2
|z| = mod(z)
integers

7. R^2 = x
i^1
Square Root
the distance from z to the origin in the complex plane
Complex Addition

8. 3
i^3
complex
imaginary
has a solution.

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

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10. 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
Rules of Complex Arithmetic
Argand diagram
Real and Imaginary Parts
z1 / z2

11. 5th. Rule of Complex Arithmetic
i^2
irrational
Liouville's Theorem -
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

12. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - sin?
ln z
subtracting complex numbers
e^(ln z)

13. (e^(-y) - e^(y)) / 2i = i sinh y
i^4
zz*
sin iy
Complex Division

14. 1
subtracting complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
e^(ln z)
i^2

15. 2ib
a + bi for some real a and b.
non-integers
Imaginary Unit
z - z*

16. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
|z-w|
How to solve (2i+3)/(9-i)
has a solution.
Integers

17. Cos n? + i sin n? (for all n integers)
subtracting complex numbers
cosh²y - sinh²y
Complex Numbers: Add & subtract
(cos? +isin?)n

18. To simplify the square root of a negative number
e^(ln z)
four different numbers: i - -i - 1 - and -1.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to multiply complex nubers(2+i)(2i-3)

19. When two complex numbers are added together.
Affix
Complex Numbers: Multiply
Complex Addition
has a solution.

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

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21. x / r
Rules of Complex Arithmetic
Polar Coordinates - cos?
Roots of Unity
Imaginary Numbers

22. y / r
Polar Coordinates - sin?
sin z
Polar Coordinates - cos?
Polar Coordinates - Multiplication by i

23. The complex number z representing a+bi.
(cos? +isin?)n
Affix
conjugate pairs
|z| = mod(z)

24. 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
'i'
zz*
the complex numbers
Liouville's Theorem -

25. All the powers of i can be written as
a + bi for some real a and b.
Subfield
four different numbers: i - -i - 1 - and -1.
Complex Division

26. Written as fractions - terminating + repeating decimals
0 if and only if a = b = 0
rational
Rules of Complex Arithmetic
z - z*

27. ½(e^(iz) + e^(-iz))
sin iy
four different numbers: i - -i - 1 - and -1.
z + z*
cos z

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
conjugate
irrational
Imaginary Unit
Rules of Complex Arithmetic

29. V(x² + y²) = |z|
i^3
Polar Coordinates - r
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. The modulus of the complex number z= a + ib now can be interpreted as
Real Numbers
We say that c+di and c-di are complex conjugates.
zz*
the distance from z to the origin in the complex plane

31. 1st. Rule of Complex Arithmetic
-1
'i'
i^2 = -1
the vector (a -b)

32. Divide moduli and subtract arguments
i^1
Polar Coordinates - Division
cos z
Real Numbers

33. (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
Subfield
the complex numbers
the vector (a -b)
How to solve (2i+3)/(9-i)

34. We can also think of the point z= a+ ib as
the vector (a -b)
Field
We say that c+di and c-di are complex conjugates.
v(-1)

35. Every complex number has the 'Standard Form':
i^2 = -1
a + bi for some real a and b.
sin z
Complex Multiplication

36. 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^3
adding complex numbers
Polar Coordinates - z?¹
Liouville's Theorem -

37. When two complex numbers are multipiled together.
Liouville's Theorem -
Complex Multiplication
complex numbers
conjugate pairs

38. 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.
non-integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power
Field

39. 1
Liouville's Theorem -
cosh²y - sinh²y
adding complex numbers
Polar Coordinates - cos?

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

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41. Like pi
cos z
transcendental
adding complex numbers
rational

42. I^2 =
non-integers
e^(ln z)
'i'
-1

43. The product of an imaginary number and its conjugate is
Complex Addition
Subfield
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division

44. (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
integers
How to multiply complex nubers(2+i)(2i-3)
'i'
Imaginary number

45. A + bi
standard form of complex numbers
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)
complex numbers

46. z1z2* / |z2|²
transcendental
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
z1 / z2

47. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a real number: (a + bi)(a - bi) = a² + b²
Absolute Value of a Complex Number
Roots of Unity
non-integers

48. When two complex numbers are subtracted from one another.
point of inflection
conjugate
Complex Subtraction
the distance from z to the origin in the complex plane

49. The square root of -1.
Imaginary Unit
Polar Coordinates - Multiplication
Complex numbers are points in the plane
Field

50. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
v(-1)
Complex Number
z + z*
multiplying complex numbers