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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.
Integers
zz*
i^4
Field

2. When two complex numbers are subtracted from one another.
point of inflection
cos iy
Complex Conjugate
Complex Subtraction

3. 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.
|z| = mod(z)
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane

4. Any number not rational
i^2 = -1
Complex Addition
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational

5. Divide moduli and subtract arguments
Polar Coordinates - Division
Polar Coordinates - z?¹
Polar Coordinates - z
Rules of Complex Arithmetic

6. I = imaginary unit - i² = -1 or i = v-1
transcendental
interchangeable
Imaginary Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

7. V(zz*) = v(a² + b²)
|z| = mod(z)
Euler's Formula
subtracting complex numbers
i^2

8. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
natural
has a solution.
Subfield
Roots of Unity

9. To simplify a complex fraction
The Complex Numbers
Polar Coordinates - Division
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.

10. 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
Polar Coordinates - z
Complex Multiplication
Integers
adding complex numbers

11. 5th. Rule of Complex Arithmetic
Field
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
De Moivre's Theorem

12. 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
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Complex Subtraction

13. Imaginary number

14. 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
Complex Numbers: Add & subtract
Every complex number has the 'Standard Form': a + bi for some real a and b.
conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two

15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
point of inflection
Complex Multiplication
interchangeable
ln z

16. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Any polynomial O(xn) - (n > 0)
interchangeable
Complex Number
Subfield

17. ½(e^(-y) +e^(y)) = cosh y
'i'
cos iy
cosh²y - sinh²y
(a + bi) = (c + bi) = (a + c) + ( b + d)i

18. 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.

19. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Euler Formula
'i'
Affix

20. x + iy = r(cos? + isin?) = re^(i?)
four different numbers: i - -i - 1 - and -1.
i^4
Integers
Polar Coordinates - z

21. y / r
zz*
Polar Coordinates - sin?
Complex Subtraction
standard form of complex numbers

22. A + bi
Irrational Number
cos z
standard form of complex numbers
z + z*

23. (a + bi) = (c + bi) =
How to add and subtract complex numbers (2-3i)-(4+6i)
The Complex Numbers
(a + c) + ( b + d)i
(cos? +isin?)n

24. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

25. Multiply moduli and add arguments
Subfield
Complex Exponentiation
Affix
Polar Coordinates - Multiplication

26. 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - sin?
four different numbers: i - -i - 1 - and -1.
i²

27. In this amazing number field every algebraic equation in z with complex coefficients
cos iy
Complex Subtraction
Complex Addition
has a solution.

28. Not on the numberline
transcendental
'i'
non-integers
The Complex Numbers

29. V(x² + y²) = |z|
zz*
Polar Coordinates - Arg(z*)
i^4
Polar Coordinates - r

30. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - r
irrational
|z-w|
Irrational Number

31. 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
Argand diagram
cos z
the complex numbers
Complex Numbers: Multiply

32. R?¹(cos? - isin?)
conjugate
point of inflection
Imaginary Unit
Polar Coordinates - z?¹

33. Like pi
transcendental
Imaginary Unit
i²
multiply the numerator and the denominator by the complex conjugate of the denominator.

34. We can also think of the point z= a+ ib as
'i'
the vector (a -b)
Complex Exponentiation
Complex Multiplication

35. 2nd. Rule of Complex Arithmetic

36. A² + b² - real and non negative
imaginary
zz*
z1 ^ (z2)
Polar Coordinates - Division

37. Derives z = a+bi
Euler Formula
non-integers
Rational Number
conjugate

38. (e^(iz) - e^(-iz)) / 2i
De Moivre's Theorem
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
sin z

39. 2a
z + z*
Argand diagram
Imaginary Unit
Complex Conjugate

40. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.
z - z*
(cos? +isin?)n

41. (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 - cos?
Complex Subtraction
Polar Coordinates - z

42. When two complex numbers are multipiled together.
Field
Roots of Unity
Complex Multiplication
conjugate pairs

43. ? = -tan?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Arg(z*)
Complex Division

44. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Roots of Unity
sin iy
Subfield

45. (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)
Subfield
Polar Coordinates - Multiplication
cos iy

46. A plot of complex numbers as points.
four different numbers: i - -i - 1 - and -1.
Argand diagram
Square Root
z1 ^ (z2)

47. The complex number z representing a+bi.
Integers
Affix
conjugate pairs
Complex Number

48. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
complex numbers
ln z
Polar Coordinates - Multiplication
multiplying complex numbers

49. The square root of -1.
How to add and subtract complex numbers (2-3i)-(4+6i)
a + bi for some real a and b.
Imaginary Unit
Real and Imaginary Parts

50. To simplify the square root of a negative number
z - z*
Polar Coordinates - Multiplication by i
Complex Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)