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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. Have radical
can't get out of the complex numbers by adding (or subtracting) or multiplying two
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
(cos? +isin?)n

2. (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
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
How to multiply complex nubers(2+i)(2i-3)
(cos? +isin?)n

3. All numbers
sin iy
For real a and b - a + bi = 0 if and only if a = b = 0
complex
z + z*

4. Root negative - has letter i
imaginary
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real and Imaginary Parts
Argand diagram

5. The modulus of the complex number z= a + ib now can be interpreted as
Integers
Real Numbers
the distance from z to the origin in the complex plane
We say that c+di and c-di are complex conjugates.

6. 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
cosh²y - sinh²y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
Rules of Complex Arithmetic

7. A subset within a field.
non-integers
Polar Coordinates - Multiplication
i^2
Subfield

8. 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 - Arg(z*)
Polar Coordinates - Multiplication
De Moivre's Theorem
We say that c+di and c-di are complex conjugates.

9. No i
Any polynomial O(xn) - (n > 0)
conjugate
a + bi for some real a and b.
real

10. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
standard form of complex numbers
Integers
z1 ^ (z2)
a real number: (a + bi)(a - bi) = a² + b²

11. Written as fractions - terminating + repeating decimals
Polar Coordinates - z
interchangeable
rational
'i'

12. Given (4-2i) the complex conjugate would be (4+2i)
non-integers
Rational Number
integers
Complex Conjugate

13. V(x² + y²) = |z|
Polar Coordinates - r
real
Complex Conjugate
Field

14. R^2 = x
Complex Numbers: Multiply
conjugate pairs
Square Root
radicals

15. A plot of complex numbers as points.
z1 / z2
complex
(cos? +isin?)n
Argand diagram

16. The product of an imaginary number and its conjugate is
adding complex numbers
e^(ln z)
a real number: (a + bi)(a - bi) = a² + b²
Affix

17. x / r
i^1
Polar Coordinates - cos?
Complex Numbers: Add & subtract
conjugate

18. z1z2* / |z2|²
i^3
Square Root
four different numbers: i - -i - 1 - and -1.
z1 / z2

19. 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
conjugate
adding complex numbers
Polar Coordinates - Arg(z*)
zz*

20. I = imaginary unit - i² = -1 or i = v-1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
The Complex Numbers
How to solve (2i+3)/(9-i)
Imaginary Numbers

21. Multiply moduli and add arguments
natural
Absolute Value of a Complex Number
Polar Coordinates - Multiplication
e^(ln z)

22. ½(e^(iz) + e^(-iz))
Polar Coordinates - sin?
cos z
complex numbers
a real number: (a + bi)(a - bi) = a² + b²

23. Rotates anticlockwise by p/2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.

24. 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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25. (e^(iz) - e^(-iz)) / 2i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i
sin z
(cos? +isin?)n

26. We can also think of the point z= a+ ib as
the vector (a -b)
We say that c+di and c-di are complex conjugates.
adding complex numbers
integers

27. 1
How to multiply complex nubers(2+i)(2i-3)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
Polar Coordinates - Multiplication by i

28. Imaginary number

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29. 2ib
z - z*
(cos? +isin?)n
Subfield
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

30. V(zz*) = v(a² + b²)
real
|z| = mod(z)
-1
Affix

31. R?¹(cos? - isin?)
'i'
Polar Coordinates - z?¹
Complex Conjugate
i^2

32. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Field
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply

33. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
z1 / z2
interchangeable
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to add and subtract complex numbers (2-3i)-(4+6i)

34. When two complex numbers are divided.
Any polynomial O(xn) - (n > 0)
Complex Division
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)

35. xpressions such as ``the complex number z'' - and ``the point z'' are now
De Moivre's Theorem
interchangeable
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number

36. 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
Rules of Complex Arithmetic
sin z
x-axis in the complex plane
Polar Coordinates - Arg(z*)

37. The square root of -1.
Integers
Imaginary Unit
(cos? +isin?)n
Every complex number has the 'Standard Form': a + bi for some real a and b.

38. A+bi
Complex Number
adding complex numbers
Polar Coordinates - cos?
Complex Number Formula

39. 1
i²
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)

40. (a + bi)(c + bi) =
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
real

41. 1st. Rule of Complex Arithmetic
i^2 = -1
adding complex numbers
conjugate pairs
point of inflection

42. Any number not rational
irrational
Polar Coordinates - z
z1 ^ (z2)
standard form of complex numbers

43. 1
i^2
i^3
|z| = mod(z)
adding complex numbers

44. 1
Argand diagram
the distance from z to the origin in the complex plane
i^0
i^3

45. A complex number may be taken to the power of another complex number.
Complex Exponentiation
has a solution.
imaginary
i^2

46. When two complex numbers are multipiled together.
Complex Multiplication
Complex Numbers: Multiply
Imaginary number
Rules of Complex Arithmetic

47. When two complex numbers are added together.
Complex Addition
Irrational Number
-1
Real and Imaginary Parts

48. A number that cannot be expressed as a fraction for any integer.
irrational
i^0
Irrational Number
Complex Numbers: Add & subtract

49. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Polar Coordinates - sin?
Imaginary Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

50. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
complex numbers
integers
adding complex numbers