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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. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract
Absolute Value of a Complex Number
i^2

2. 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
Complex numbers are points in the plane
the vector (a -b)
has a solution.
the complex numbers

3. Derives z = a+bi
the complex numbers
Imaginary number
Euler Formula
Complex Subtraction

4. Numbers on a numberline
integers
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. 1
Imaginary Numbers
i²
Subfield
i^4

6. Divide moduli and subtract arguments
complex numbers
Polar Coordinates - Division
z - z*
Complex Numbers: Multiply

7. 2ib
z - z*
a real number: (a + bi)(a - bi) = a² + b²
Real and Imaginary Parts
standard form of complex numbers

8. A number that cannot be expressed as a fraction for any integer.
standard form of complex numbers
conjugate
Irrational Number
Polar Coordinates - Arg(z*)

9. Any number not rational
|z| = mod(z)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - r
irrational

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 Number
Rules of Complex Arithmetic
Complex Numbers: Multiply
standard form of complex numbers

11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
z - z*
Complex Number Formula
point of inflection
Integers

12. Every complex number has the 'Standard Form':
Every complex number has the 'Standard Form': a + bi for some real a and b.
subtracting complex numbers
a + bi for some real a and b.
Complex Numbers: Add & subtract

13. x / r
radicals
Polar Coordinates - cos?
Polar Coordinates - Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

14. When two complex numbers are subtracted from one another.
Complex Subtraction
i^3
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number

15. Given (4-2i) the complex conjugate would be (4+2i)
integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
Complex Conjugate

16. Multiply moduli and add arguments
z + z*
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication
the distance from z to the origin in the complex plane

17. Imaginary number

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18. A complex number and its conjugate
Polar Coordinates - Multiplication
i^2
i^2 = -1
conjugate pairs

19. 4th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply
(a + bi) = (c + bi) = (a + c) + ( b + d)i
-1

20. 1st. Rule of Complex Arithmetic
z - z*
zz*
four different numbers: i - -i - 1 - and -1.
i^2 = -1

21. 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 Number
How to add and subtract complex numbers (2-3i)-(4+6i)
The Complex Numbers
Complex numbers are points in the plane

22. (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
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹
Subfield
Integers

23. ½(e^(iz) + e^(-iz))
multiply the numerator and the denominator by the complex conjugate of the denominator.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Absolute Value of a Complex Number
cos z

24. ? = -tan?
The Complex Numbers
Polar Coordinates - Arg(z*)
i^3
We say that c+di and c-di are complex conjugates.

25. 2nd. Rule of Complex Arithmetic

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26. 1
|z-w|
i^2
conjugate
Rational Number

27. A+bi
the vector (a -b)
'i'
Complex Numbers: Multiply
Complex Number Formula

28. For real a and b - a + bi =
Polar Coordinates - Multiplication by i
point of inflection
Polar Coordinates - Multiplication
0 if and only if a = b = 0

29. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
a real number: (a + bi)(a - bi) = a² + b²
How to solve (2i+3)/(9-i)
Complex Subtraction

30. 5th. Rule of Complex Arithmetic
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
Polar Coordinates - z?¹

31. 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^3
i^0
Affix

32. When two complex numbers are added together.
Real and Imaginary Parts
Complex Addition
conjugate pairs
integers

33. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
conjugate
(cos? +isin?)n
Polar Coordinates - z?¹

34. A subset within a field.
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield
Complex Number Formula

35. A complex number may be taken to the power of another complex number.
Complex Exponentiation
We say that c+di and c-di are complex conjugates.
(cos? +isin?)n
i^0

36. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Roots of Unity
Complex Multiplication
subtracting complex numbers
|z| = mod(z)

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

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38. z1z2* / |z2|²
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z1 / z2
Complex Subtraction
Complex Multiplication

39. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y
Polar Coordinates - Multiplication by i
Complex Addition

40. y / r
We say that c+di and c-di are complex conjugates.
conjugate pairs
Polar Coordinates - sin?
Complex Exponentiation

41. Real and imaginary numbers
Complex Numbers: Add & subtract
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
complex numbers

42. The complex number z representing a+bi.
Irrational Number
Affix
z1 / z2
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

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

44. x + iy = r(cos? + isin?) = re^(i?)
The Complex Numbers
Polar Coordinates - z
Integers
subtracting complex numbers

45. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex
Rational Number
Liouville's Theorem -
transcendental

46. The field of all rational and irrational numbers.
De Moivre's Theorem
standard form of complex numbers
Real Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. Have radical
Every complex number has the 'Standard Form': a + bi for some real a and b.
(cos? +isin?)n
sin iy
radicals

48. We can also think of the point z= a+ ib as
the vector (a -b)
Complex Division
|z-w|
adding complex numbers

49. E ^ (z2 ln z1)
transcendental
cos iy
Rules of Complex Arithmetic
z1 ^ (z2)

50. V(zz*) = v(a² + b²)
Irrational Number
|z| = mod(z)
conjugate pairs
Field