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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. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
complex numbers
complex
(a + c) + ( b + d)i
Integers

2. (a + bi) = (c + bi) =
z1 ^ (z2)
Complex Subtraction
Complex Conjugate
(a + c) + ( b + d)i

3. 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
real
natural
0 if and only if a = b = 0
the complex numbers

4. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Division
i²
Rules of Complex Arithmetic

5. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.
interchangeable

6. Numbers on a numberline
integers
0 if and only if a = b = 0
irrational
Any polynomial O(xn) - (n > 0)

7. Derives z = a+bi
Euler's Formula
Subfield
Imaginary Numbers
Euler Formula

8. 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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9. V(zz*) = v(a² + b²)
four different numbers: i - -i - 1 - and -1.
Imaginary Unit
|z| = mod(z)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

10. 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 Numbers
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

11. R?¹(cos? - isin?)
i^4
Affix
Polar Coordinates - z?¹
cosh²y - sinh²y

12. E^(ln r) e^(i?) e^(2pin)
the complex numbers
e^(ln z)
-1
cosh²y - sinh²y

13. 3rd. Rule of Complex Arithmetic
Complex Subtraction
Euler's Formula
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)

14. We can also think of the point z= a+ ib as
the vector (a -b)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers
integers

15. A² + b² - real and non negative
the vector (a -b)
sin z
zz*
Rules of Complex Arithmetic

16. 1
i²
multiplying complex numbers
(cos? +isin?)n
i^2 = -1

17. The complex number z representing a+bi.
x-axis in the complex plane
'i'
Affix
Polar Coordinates - Multiplication by i

18. A subset within a field.
four different numbers: i - -i - 1 - and -1.
How to find any Power
Subfield
subtracting complex numbers

19. 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
Complex Number Formula
Euler's Formula
the complex numbers

20. Written as fractions - terminating + repeating decimals
rational
Imaginary number
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i

21. The square root of -1.
The Complex Numbers
subtracting complex numbers
Imaginary Unit
Polar Coordinates - r

22. A number that cannot be expressed as a fraction for any integer.
Irrational Number
x-axis in the complex plane
real
sin iy

23. Like pi
a + bi for some real a and b.
transcendental
Subfield
conjugate pairs

24. The field of all rational and irrational numbers.
Complex Division
Complex Number
Real Numbers
transcendental

25. 4th. Rule of Complex Arithmetic
conjugate pairs
Rules of Complex Arithmetic
radicals
(a + bi) = (c + bi) = (a + c) + ( b + d)i

26. Cos n? + i sin n? (for all n integers)
z1 / z2
(cos? +isin?)n
i^2
The Complex Numbers

27. 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
Euler Formula
Real and Imaginary Parts
z1 ^ (z2)
subtracting complex numbers

28. When two complex numbers are added together.
|z| = mod(z)
Absolute Value of a Complex Number
Complex Addition
Complex Numbers: Multiply

29. I = imaginary unit - i² = -1 or i = v-1
Subfield
Imaginary Numbers
Field
Every complex number has the 'Standard Form': a + bi for some real a and b.

30. x / r
x-axis in the complex plane
Polar Coordinates - cos?
Complex numbers are points in the plane
Imaginary Unit

31. A + bi
Polar Coordinates - r
z1 ^ (z2)
standard form of complex numbers
imaginary

32. 2ib
z - z*
Square Root
Field
|z| = mod(z)

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
i^0
zz*
Polar Coordinates - Multiplication
How to solve (2i+3)/(9-i)

34. Multiply moduli and add arguments
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication
Complex Number Formula
multiplying complex numbers

35. I
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
v(-1)
the distance from z to the origin in the complex plane

36. I^2 =
cosh²y - sinh²y
-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts

37. R^2 = x
Complex Division
a + bi for some real a and b.
e^(ln z)
Square Root

38. When two complex numbers are multipiled together.
Irrational Number
has a solution.
Complex Multiplication
cos z

39. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Real and Imaginary Parts
Imaginary Numbers
irrational

40. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rational Number
interchangeable
Roots of Unity

41. Equivalent to an Imaginary Unit.
irrational
point of inflection
conjugate pairs
Imaginary number

42. For real a and b - a + bi =
(cos? +isin?)n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0

43. ½(e^(iz) + e^(-iz))
the distance from z to the origin in the complex plane
The Complex Numbers
cos z
Real Numbers

44. When two complex numbers are subtracted from one another.
Complex Addition
Complex Subtraction
adding complex numbers
Imaginary number

45. Divide moduli and subtract arguments
complex numbers
Complex Numbers: Add & subtract
Polar Coordinates - Division
irrational

46. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
i^4
Irrational Number
i^2 = -1

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

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48. 1
'i'
i^2
integers
For real a and b - a + bi = 0 if and only if a = b = 0

49. All numbers
Polar Coordinates - Multiplication by i
The Complex Numbers
complex
Complex Subtraction

50. A complex number may be taken to the power of another complex number.
Complex Exponentiation
complex numbers
Imaginary Numbers
complex