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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. Not on the numberline
Affix
non-integers
zz*
four different numbers: i - -i - 1 - and -1.

2. 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
Polar Coordinates - sin?
We say that c+di and c-di are complex conjugates.
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0

3. 4th. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
Complex Subtraction
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i

4. All numbers
complex
Polar Coordinates - Multiplication by i
De Moivre's Theorem
The Complex Numbers

5. 2ib
Complex Addition
Real and Imaginary Parts
z - z*
conjugate

6. Written as fractions - terminating + repeating decimals
For real a and b - a + bi = 0 if and only if a = b = 0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
De Moivre's Theorem

7. R^2 = x
Square Root
multiplying complex numbers
(a + c) + ( b + d)i
Affix

8. 1
Polar Coordinates - Multiplication
i^0
multiplying complex numbers
natural

9. 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.
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Addition
imaginary
Complex numbers are points in the plane

10. xpressions such as ``the complex number z'' - and ``the point z'' are now
|z| = mod(z)
sin iy
interchangeable
Polar Coordinates - Division

11. A number that cannot be expressed as a fraction for any integer.
Irrational Number
complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers

12. Imaginary number

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13. The modulus of the complex number z= a + ib now can be interpreted as
i^1
Integers
Polar Coordinates - z?¹
the distance from z to the origin in the complex plane

14. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
cos iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two
conjugate pairs

15. ½(e^(-y) +e^(y)) = cosh y
a + bi for some real a and b.
the vector (a -b)
z + z*
cos iy

16. 1
i^2
Square Root
z1 / z2
i^1

17. 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
cosh²y - sinh²y
i^2 = -1
complex

18. 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
Imaginary Unit
x-axis in the complex plane
Real Numbers

19. Numbers on a numberline
can't get out of the complex numbers by adding (or subtracting) or multiplying two
integers
Complex Conjugate
i²

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
How to multiply complex nubers(2+i)(2i-3)
x-axis in the complex plane
Polar Coordinates - z?¹
Imaginary number

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

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22. 1
point of inflection
We say that c+di and c-di are complex conjugates.
cosh²y - sinh²y
a real number: (a + bi)(a - bi) = a² + b²

23. y / r
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - sin?
Euler Formula
Complex Conjugate

24. V(zz*) = v(a² + b²)
Complex Conjugate
e^(ln z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)

25. (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)
De Moivre's Theorem
Integers
i^1

26. In this amazing number field every algebraic equation in z with complex coefficients
a real number: (a + bi)(a - bi) = a² + b²
How to add and subtract complex numbers (2-3i)-(4+6i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.

27. (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
sin iy
How to solve (2i+3)/(9-i)
i^2
ln z

28. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
non-integers
conjugate
complex
standard form of complex numbers

29. To simplify the square root of a negative number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals
z + z*

30. 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
Argand diagram
adding complex numbers
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z

31. Derives z = a+bi
Euler Formula
Polar Coordinates - Division
sin iy
Real and Imaginary Parts

32. Starts at 1 - does not include 0
We say that c+di and c-di are complex conjugates.
natural
Absolute Value of a Complex Number
transcendental

33. The square root of -1.
|z| = mod(z)
How to find any Power
Imaginary Unit
Euler Formula

34. When two complex numbers are divided.
Complex Division
standard form of complex numbers
i^4
sin z

35. z1z2* / |z2|²
i^4
cos z
subtracting complex numbers
z1 / z2

36. Multiply moduli and add arguments
De Moivre's Theorem
the complex numbers
real
Polar Coordinates - Multiplication

37. 1
i²
Complex Number Formula
Polar Coordinates - Multiplication
Complex Addition

38. Like pi
Liouville's Theorem -
conjugate
transcendental
i^3

39. E^(ln r) e^(i?) e^(2pin)
i^4
a real number: (a + bi)(a - bi) = a² + b²
e^(ln z)
i^3

40. The complex number z representing a+bi.
0 if and only if a = b = 0
Affix
multiplying complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

41. To simplify a complex fraction
Complex Number
sin z
the complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

42. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Irrational Number
Argand diagram
Polar Coordinates - z
Field

43. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
z1 ^ (z2)
Subfield
Complex numbers are points in the plane
ln z

44. For real a and b - a + bi =
0 if and only if a = b = 0
Complex Exponentiation
i^1
Rational Number

45. Given (4-2i) the complex conjugate would be (4+2i)
Integers
Complex Conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.
Field

46. A² + b² - real and non negative
zz*
sin iy
cos z
Polar Coordinates - z?¹

47. 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
(a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)
Real and Imaginary Parts
-1

48. Real and imaginary numbers
Polar Coordinates - sin?
complex numbers
Complex Conjugate
rational

49. (e^(iz) - e^(-iz)) / 2i
sin z
Absolute Value of a Complex Number
cos iy
a + bi for some real a and b.

50. 2nd. Rule of Complex Arithmetic

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