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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. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Complex Addition
Polar Coordinates - Arg(z*)
non-integers

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
cos z
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Integers

3. Cos n? + i sin n? (for all n integers)
subtracting complex numbers
Any polynomial O(xn) - (n > 0)
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

4. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Square Root
De Moivre's Theorem
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

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

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6. 3rd. Rule of Complex Arithmetic
rational
For real a and b - a + bi = 0 if and only if a = b = 0
v(-1)
Imaginary Numbers

7. ½(e^(iz) + e^(-iz))
interchangeable
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the vector (a -b)
cos z

8. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
imaginary
integers
adding complex numbers

9. 1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract
i^4
Real Numbers

10. z1z2* / |z2|²
How to solve (2i+3)/(9-i)
Roots of Unity
Complex Number Formula
z1 / z2

11. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
(cos? +isin?)n
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z
multiplying complex numbers

12. V(x² + y²) = |z|
Absolute Value of a Complex Number
How to solve (2i+3)/(9-i)
Polar Coordinates - r
Polar Coordinates - sin?

13. The field of all rational and irrational numbers.
i^2
v(-1)
imaginary
Real Numbers

14. Not on the numberline
z1 / z2
How to multiply complex nubers(2+i)(2i-3)
non-integers
conjugate

15. 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
subtracting complex numbers
Complex Addition
Real and Imaginary Parts
i²

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - sin?
Roots of Unity
v(-1)
integers

17. 4th. Rule of Complex Arithmetic
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix

18. When two complex numbers are divided.
Complex Division
|z-w|
Subfield
Complex Multiplication

19. A² + b² - real and non negative
cos z
|z-w|
cosh²y - sinh²y
zz*

20. When two complex numbers are subtracted from one another.
Polar Coordinates - Division
Argand diagram
(cos? +isin?)n
Complex Subtraction

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

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22. 3
i^3
Complex Exponentiation
Complex Addition
Polar Coordinates - Arg(z*)

23. A number that cannot be expressed as a fraction for any integer.
multiplying complex numbers
Polar Coordinates - Multiplication
z1 / z2
Irrational Number

24. To simplify a complex fraction
We say that c+di and c-di are complex conjugates.
standard form of complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Exponentiation

25. Starts at 1 - does not include 0
natural
conjugate pairs
i^4
Complex Multiplication

26. 1st. Rule of Complex Arithmetic
Rational Number
i^2 = -1
Polar Coordinates - z?¹
z1 ^ (z2)

27. A+bi
How to add and subtract complex numbers (2-3i)-(4+6i)
integers
Complex Number Formula
|z-w|

28. A subset within a field.
Complex Numbers: Multiply
How to find any Power
Subfield
Field

29. R?¹(cos? - isin?)
Polar Coordinates - cos?
Polar Coordinates - z?¹
i^4
|z| = mod(z)

30. I = imaginary unit - i² = -1 or i = v-1
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers
natural
rational

31. R^2 = x
Integers
Square Root
irrational
The Complex Numbers

32. Every complex number has the 'Standard Form':
point of inflection
Integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a + bi for some real a and b.

33. 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
Complex Numbers: Add & subtract
a + bi for some real a and b.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

34. (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
complex
How to multiply complex nubers(2+i)(2i-3)
z1 / z2
Square Root

35. All numbers
Square Root
i^2
i²
complex

36. Root negative - has letter i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
imaginary
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)

37. 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
Rational Number
radicals
Affix
Complex Numbers: Add & subtract

38. Equivalent to an Imaginary Unit.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number
Polar Coordinates - r
How to multiply complex nubers(2+i)(2i-3)

39. Where the curvature of the graph changes
point of inflection
Complex Number Formula
x-axis in the complex plane
Imaginary Unit

40. We see in this way that the distance between two points z and w in the complex plane is
ln z
Polar Coordinates - Multiplication
Liouville's Theorem -
|z-w|

41. 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
integers
the vector (a -b)
adding complex numbers
Rules of Complex Arithmetic

42. Has exactly n roots by the fundamental theorem of algebra
radicals
Polar Coordinates - cos?
the distance from z to the origin in the complex plane
Any polynomial O(xn) - (n > 0)

43. 2nd. Rule of Complex Arithmetic

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44. x / r
non-integers
the vector (a -b)
Polar Coordinates - cos?
Subfield

45. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + c) + ( b + d)i
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane

46. Rotates anticlockwise by p/2
real
i^1
Polar Coordinates - Multiplication by i
i^3

47. Numbers on a numberline
integers
Polar Coordinates - cos?
Real Numbers
Subfield

48. Like pi
a real number: (a + bi)(a - bi) = a² + b²
the complex numbers
e^(ln z)
transcendental

49. The modulus of the complex number z= a + ib now can be interpreted as
Affix
the distance from z to the origin in the complex plane
real
Euler's Formula

50. y / r
Irrational Number
Complex Number
Polar Coordinates - sin?
z - z*