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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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1. 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
0 if and only if a = b = 0
Rules of Complex Arithmetic
rational
Liouville's Theorem -

2. 4th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.

3. 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
We say that c+di and c-di are complex conjugates.
Roots of Unity
Complex Division
v(-1)

4. 1
Polar Coordinates - Multiplication
Rules of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
i^0

5. We can also think of the point z= a+ ib as
the vector (a -b)
How to solve (2i+3)/(9-i)
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2

6. y / r
Complex Division
irrational
Complex Numbers: Add & subtract
Polar Coordinates - sin?

7. Starts at 1 - does not include 0
natural
|z| = mod(z)
Polar Coordinates - z?¹
Roots of Unity

8. The square root of -1.
Imaginary Unit
Square Root
i^3
imaginary

9. x + iy = r(cos? + isin?) = re^(i?)
Complex Number Formula
e^(ln z)
i^3
Polar Coordinates - z

10. Written as fractions - terminating + repeating decimals
interchangeable
Complex Multiplication
irrational
rational

11. The reals are just the
radicals
x-axis in the complex plane
-1
complex numbers

12. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
'i'
a + bi for some real a and b.
cosh²y - sinh²y
Absolute Value of a Complex Number

13. No i
real
conjugate pairs
De Moivre's Theorem
i²

14. Real and imaginary numbers
conjugate pairs
Any polynomial O(xn) - (n > 0)
complex numbers
x-axis in the complex plane

15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary number
cosh²y - sinh²y
Field
z1 ^ (z2)

16. Any number not rational
i^3
irrational
(cos? +isin?)n
standard form of complex numbers

17. ½(e^(iz) + e^(-iz))
point of inflection
conjugate pairs
cos z
sin z

18. I^2 =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
-1
Real Numbers

19. Cos n? + i sin n? (for all n integers)
the distance from z to the origin in the complex plane
Complex Multiplication
Field
(cos? +isin?)n

20. 1st. Rule of Complex Arithmetic
point of inflection
i^2
i^2 = -1
conjugate

21. ? = -tan?
Polar Coordinates - r
The Complex Numbers
Polar Coordinates - Arg(z*)
Subfield

22. Derives z = a+bi
sin z
Any polynomial O(xn) - (n > 0)
a real number: (a + bi)(a - bi) = a² + b²
Euler Formula

23. 3
cos iy
ln z
|z| = mod(z)
i^3

24. Divide moduli and subtract arguments
How to solve (2i+3)/(9-i)
Polar Coordinates - Division
cosh²y - sinh²y
Absolute Value of a Complex Number

25. V(x² + y²) = |z|
Polar Coordinates - r
cos iy
complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. We see in this way that the distance between two points z and w in the complex plane is
How to multiply complex nubers(2+i)(2i-3)
irrational
|z-w|
|z| = mod(z)

27. 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.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
cos iy
Complex numbers are points in the plane

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

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29. 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
the complex numbers
ln z
Complex Multiplication
Roots of Unity

30. A subset within a field.
How to multiply complex nubers(2+i)(2i-3)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Subfield
a real number: (a + bi)(a - bi) = a² + b²

31. 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
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)
How to solve (2i+3)/(9-i)
Complex Exponentiation

32. To simplify a complex fraction
Euler Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Multiplication
0 if and only if a = b = 0

33. To simplify the square root of a negative number
Real and Imaginary Parts
How to find any Power
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

34. 2a
z + z*
-1
has a solution.
adding complex numbers

35. 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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36. I
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2
How to add and subtract complex numbers (2-3i)-(4+6i)

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

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38. A complex number and its conjugate
conjugate pairs
a real number: (a + bi)(a - bi) = a² + b²
Subfield
ln z

39. 1
cos iy
Polar Coordinates - cos?
i^2
Rational Number

40. A plot of complex numbers as points.
natural
the vector (a -b)
z1 / z2
Argand diagram

41. (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)
i^2
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Division

42. (a + bi) = (c + bi) =
Rational Number
(a + c) + ( b + d)i
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0

43. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
cos z
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number Formula

44. A² + b² - real and non negative
Polar Coordinates - Multiplication by i
natural
Roots of Unity
zz*

45. All numbers
Affix
the vector (a -b)
complex
Complex Number

46. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - r
interchangeable
De Moivre's Theorem
Polar Coordinates - Multiplication by i

47. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^4
Imaginary number
x-axis in the complex plane
multiplying complex numbers

48. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
v(-1)
Real Numbers
i^4

49. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
non-integers
cos z
Integers
Complex Numbers: Multiply

50. 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
natural
adding complex numbers
i^1
Polar Coordinates - Division