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

2. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Conjugate
The Complex Numbers

3. 2ib
Euler's Formula
z - z*
(a + c) + ( b + d)i
real

4. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
conjugate pairs
z - z*
Roots of Unity
cos z

5. A complex number may be taken to the power of another complex number.
Complex Exponentiation
sin iy
ln z
Subfield

6. 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
Roots of Unity
z + z*
Real and Imaginary Parts
Real Numbers

7. The reals are just the
radicals
How to multiply complex nubers(2+i)(2i-3)
non-integers
x-axis in the complex plane

8. 1
|z| = mod(z)
Polar Coordinates - cos?
v(-1)
i^4

9. x / r
Argand diagram
Polar Coordinates - cos?
adding complex numbers
i^2

10. E^(ln r) e^(i?) e^(2pin)
has a solution.
e^(ln z)
Absolute Value of a Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.

11. 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.
Complex Numbers: Multiply
Real and Imaginary Parts
the vector (a -b)
conjugate pairs

12. 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
cos z
Absolute Value of a Complex Number
sin z

13. Cos n? + i sin n? (for all n integers)
The Complex Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(cos? +isin?)n
i^2 = -1

14. All the powers of i can be written as
(cos? +isin?)n
Polar Coordinates - z?¹
We say that c+di and c-di are complex conjugates.
four different numbers: i - -i - 1 - and -1.

15. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
ln z
i²
Absolute Value of a Complex Number
integers

16. R?¹(cos? - isin?)
integers
Subfield
Polar Coordinates - z?¹
e^(ln z)

17. 2a
(a + c) + ( b + d)i
'i'
z - z*
z + z*

18. Numbers on a numberline
irrational
subtracting complex numbers
integers
Complex Multiplication

19. Have radical
radicals
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary number

20. (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
cosh²y - sinh²y
ln z
How to multiply complex nubers(2+i)(2i-3)
How to solve (2i+3)/(9-i)

21. We see in this way that the distance between two points z and w in the complex plane is
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z-w|
transcendental
Polar Coordinates - Division

22. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
irrational
Irrational Number
How to add and subtract complex numbers (2-3i)-(4+6i)
adding complex numbers

23. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
z - z*
a + bi for some real a and b.
i^0

24. When two complex numbers are multipiled together.
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication
point of inflection
subtracting complex numbers

25. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
the vector (a -b)
Polar Coordinates - z?¹
Polar Coordinates - r

26. 1
Complex Numbers: Multiply
z1 ^ (z2)
Rules of Complex Arithmetic
i^0

27. V(zz*) = v(a² + b²)
Complex Numbers: Multiply
Complex numbers are points in the plane
|z| = mod(z)
zz*

28. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
the distance from z to the origin in the complex plane
cos z
Complex Number

29. Given (4-2i) the complex conjugate would be (4+2i)
z - z*
cosh²y - sinh²y
Complex Conjugate
Rules of Complex Arithmetic

30. The square root of -1.
Imaginary Unit
i^3
Imaginary Numbers
i²

31. 1st. Rule of Complex Arithmetic
Liouville's Theorem -
i^4
Complex Multiplication
i^2 = -1

32. 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.
How to find any Power
-1
(cos? +isin?)n
Complex Conjugate

33. The field of all rational and irrational numbers.
the complex numbers
Real Numbers
Polar Coordinates - z?¹
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

34. Starts at 1 - does not include 0
Complex Conjugate
Imaginary Numbers
natural
Roots of Unity

35. Not on the numberline
ln z
cosh²y - sinh²y
Polar Coordinates - z?¹
non-integers

36. 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.

37. Where the curvature of the graph changes
Imaginary number
standard form of complex numbers
point of inflection
Real and Imaginary Parts

38. 1
Complex Exponentiation
Complex Addition
cosh²y - sinh²y
a + bi for some real a and b.

39. 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
Field
Rules of Complex Arithmetic
Polar Coordinates - cos?
adding complex numbers

40. 2nd. Rule of Complex Arithmetic

41. y / r
Polar Coordinates - sin?
Euler Formula
We say that c+di and c-di are complex conjugates.
Complex Number Formula

42. Real and imaginary numbers
conjugate
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

43. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
We say that c+di and c-di are complex conjugates.
Integers
Complex Conjugate
complex

44. A+bi
'i'
Complex Number Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)

45. 1
Argand diagram
i²
Roots of Unity
the complex numbers

46. A number that cannot be expressed as a fraction for any integer.
Irrational Number
four different numbers: i - -i - 1 - and -1.
|z-w|
(a + bi) = (c + bi) = (a + c) + ( b + d)i

47. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin z
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. Has exactly n roots by the fundamental theorem of algebra
zz*
0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division

49. When two complex numbers are subtracted from one another.
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹

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