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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.
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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. 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
For real a and b - a + bi = 0 if and only if a = b = 0
adding complex numbers
|z-w|
|z| = mod(z)

2. 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 pairs
z1 ^ (z2)
natural

3. Any number not rational
How to add and subtract complex numbers (2-3i)-(4+6i)
zz*
irrational
Square Root

4. 5th. Rule of Complex Arithmetic
-1
Polar Coordinates - z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)

5. 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
Polar Coordinates - sin?
Field
i^2

6. Starts at 1 - does not include 0
i^3
Polar Coordinates - z
Imaginary Numbers
natural

7. 1
Complex Numbers: Multiply
i²
Field
i^2 = -1

8. (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
Euler's Formula
interchangeable
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)

9. The product of an imaginary number and its conjugate is
natural
a real number: (a + bi)(a - bi) = a² + b²
Integers
Absolute Value of a Complex Number

10. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
i^0
interchangeable
Argand diagram

11. R^2 = x
rational
(cos? +isin?)n
Square Root
integers

12. V(zz*) = v(a² + b²)
0 if and only if a = b = 0
|z| = mod(z)
Polar Coordinates - Division
i²

13. E ^ (z2 ln z1)
i^2
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Numbers: Multiply
z1 ^ (z2)

14. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Complex Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Number

15. Written as fractions - terminating + repeating decimals
rational
imaginary
i^1
Euler's Formula

16. z1z2* / |z2|²
four different numbers: i - -i - 1 - and -1.
z1 / z2
'i'
Imaginary Unit

17. 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
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Real and Imaginary Parts

18. 1
non-integers
conjugate
cosh²y - sinh²y
i^0

19. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Polar Coordinates - Division
conjugate
v(-1)
interchangeable

20. Root negative - has letter i
Euler Formula
v(-1)
-1
imaginary

21. ½(e^(iz) + e^(-iz))
Absolute Value of a Complex Number
Polar Coordinates - sin?
De Moivre's Theorem
cos z

22. Where the curvature of the graph changes
Affix
Rules of Complex Arithmetic
point of inflection
How to find any Power

23. 1
conjugate
|z| = mod(z)
i^0
zz*

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

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

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26. 4th. Rule of Complex Arithmetic
Square Root
(cos? +isin?)n
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i

27. A + bi
|z| = mod(z)
Polar Coordinates - cos?
standard form of complex numbers
adding complex numbers

28. When two complex numbers are multipiled together.
How to solve (2i+3)/(9-i)
Complex Multiplication
Argand diagram
Roots of Unity

29. The square root of -1.
We say that c+di and c-di are complex conjugates.
Imaginary Unit
Integers
Polar Coordinates - z

30. A² + b² - real and non negative
Roots of Unity
The Complex Numbers
zz*
radicals

31. I
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
has a solution.
i^1

32. y / r
z - z*
interchangeable
Complex Multiplication
Polar Coordinates - sin?

33. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
multiplying complex numbers
Complex Numbers: Multiply
Argand diagram

34. A+bi
Integers
Complex Number Formula
i^1
transcendental

35. (e^(iz) - e^(-iz)) / 2i
four different numbers: i - -i - 1 - and -1.
sin z
complex
Polar Coordinates - sin?

36. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Rational Number
|z| = mod(z)
Absolute Value of a Complex Number
Argand diagram

37. 3
i^3
0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental

38. Real and imaginary numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
complex numbers
Field
has a solution.

39. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^3
ln z
standard form of complex numbers

40. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication by i
v(-1)
|z| = mod(z)

41. Every complex number has the 'Standard Form':
the vector (a -b)
a + bi for some real a and b.
Polar Coordinates - Multiplication by i
transcendental

42. A plot of complex numbers as points.
Argand diagram
sin iy
Polar Coordinates - cos?
(a + bi) = (c + bi) = (a + c) + ( b + d)i

43. (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
(cos? +isin?)n
How to solve (2i+3)/(9-i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cos z

44. 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.
complex numbers
Euler Formula
radicals
Complex numbers are points in the plane

45. We can also think of the point z= a+ ib as
Polar Coordinates - Multiplication
ln z
the vector (a -b)
z + z*

46. All numbers
Imaginary Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cosh²y - sinh²y
complex

47. 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
Rational Number
We say that c+di and c-di are complex conjugates.
z + z*
Polar Coordinates - z?¹

48. Multiply moduli and add arguments
Complex Exponentiation
Polar Coordinates - Multiplication
adding complex numbers
x-axis in the complex plane

49. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers
z1 / z2
We say that c+di and c-di are complex conjugates.

50. 1st. Rule of Complex Arithmetic
Imaginary Unit
i^2 = -1
Euler Formula
standard form of complex numbers