Math symbol meaning question?? pls help. = )?

Question

ok. now in a math question, sometimes you will read this... "where a,b,and c € R" . (the R is really bold) or you will see a sequence and it will say at the end " where x € Z (the Z is really bold " ok, well what does all that mean? the whole "€ R/ Z/ N " is it important? what should i look for in a math problem when i am given that info.

ex. P(z)= Z^3 + aZ^2 + bZ + c, where a, b, and c € R. two of the roots of P(z) = 0 are -2 and ( -3 + 2i ). Find the value of a, of b, and of c.

see, like that prob. so what would i look for? hmm...

oh, and if anyone knows the answer to the ex. math prob. i gave, lol, can you help me with it? i dont know the answer . thanx.

Answer 1

The Z is the set of all integers.
The R is the set of all real numbers.

N - natural numbers
Q - rational numbers
C - complex numbers

€ - "is/are in"

It is important, though usually ignored in basic problems.

Your problem:
P(z)= Z^3 + aZ^2 + bZ + c, where a, b, and c € R. two of the roots of P(z) = 0 are -2 and ( -3 + 2i ).

a, b, and c € R
^ This tells you that a,b, and c are all in the set of real numbers

0 = (-2^3) + a(-2^2) + b(-2) + c
0 = -8 + 4a - 2b + c

0 = (-3 + 2i)^3 + a(-3 + 2i)^2 + b(-3 + 2i) + c
(-3 + 2i)^2 = 9 - 12i + 4i^2 = 5 - 12i
(-3 + 2i)^3 = (5 - 12i)(-3 + 2i) = -15 + 46i - 24i^2 = 9 + 46i
0 = (9 + 46i) + a(5 - 12i) + b(-3 + 2i) + c
0 = 9 + 46i + 5a - 12ai - 3b + 2bi + c
0 = 9 + 5a - 3b + 46i - 12ai + 2bi + c
0 = 9 + 5a - 3b + c + i(46 - 12a + 2b)
We are looking for real numbers only so:
46 - 12a + 2b = 0

Therefore:
9 + 5a - 3b + c = 0
And our first equation was:
0 = -8 + 4a - 2b + c

So here is your system:
-12a + 2b = -46
5a - 3b + c = -9
4a - 2b + c = -8

When you solve it, you should get:
a = -24/5
b = -29/5
c = 78/5

Answer 2

Wanted to find a reference so you would have something to refer to later if you need to.

R is the set of real numbers. this includes all rational numbers (those that can be written as a fraction such as 2, 0.5, etc) and those that can't (pi, SQRT(3), etc)
N is the set of natural numbers {1,2,3,...}
Z is the set of integers { ..... -2,-1,0,1,2,...}

Use the roots and add in the third in a general form (make it imaginary since the third root might be imaginary):
(z + 2)(z + 3 - 2i)(z +m + in) = z^3 + az^2 + bz + c = 0

expand the left hand side:
(z^2 + 3z - 2iz + 2z + 6 - 4i)(z + m + in)
(z^2 - 2iz + 5z - 4i + 6)(z + m + in)
z^3 - 2iz^2 + 5z^2 - 4iz + 6z + mz^2 - 2imz + 5mz - 4im + 6m + inz^2 + 2nz + 5inz + 4n + 6ni

The z^3 can be eliminated from each side.

Look at c first: 4n + 6ni - 4im + 6m = c
4n + 6m + i(6n - 4m) = c
since c is a real number, the imaginary part must be 0 so:
6n - 4m = 0 and 3n = 2m and c = 13n

Look at the z term: -4iz + 6z + 5mz - 2imz + 2nz + 5inz
z[(6 + 5m + 2n) + i(5n - 2m- 4)] = bz
Again imaginary part must be 0 so: 5n - 2m - 4 = 0
Combine with above: 5n - 3n - 4 = 0 so 2n = 4
And: n = 2 and m = 3 giving, from above, c = 26

z[(6 + 5m + 2n)] = (6 + 15 + 4)z = 25z and b = 25

Look at the z^2 term: - 2iz^2 + 5z^2 + mz^2 + inz^2
z^2[(5 + m) + i(n - 2)] = az^2
Again the imaginary part must be 0 so this gives n = 2 which agrees with the previous calculation.
"a" is 5 + m so: a = 8

a = 8, b = 25, c = 26

Answer 3

The symbol € is just telling you what set of numbers the variables belong to. For example, the "bold R" stands for real numbers, so the phrase "where a, b, and c € R" literally means that the variables a, b, and c are real numbers. The other bold letters are meanings for other sets, such as "Z" for the set of all integers and "N" for the set of all natural numbers.

For the complete set of special sets, you can look here:
http://en.wikipedia.org/wiki/Set#Special_sets

Answer 4

where a,b,and c € R
can be read as
a, b and c are elements of the set of all real numbers, R

€ is the symbol to mean "an elemnt of" used in set theory

R is the set of all real numbers

Z is the set of all integers, ... -2, -1, 0, 1, 2, ...

N is the set of all natural numbers, 0, 1, 2 ...

Answer 5

the "€" sign means "belongs to" in layman's terms. The bold R means the real numbers. ( http://en.wikipedia.org/wiki/Real_number ) There are other types of "bolded" letters too, and they represent different things. Here is a table of all the letters ( http://en.wikipedia.org/wiki/Blackboard_bold )

As you see, Z means integers and N means natural numbers. So putting it all together, a, b, and c € R just means "a, b, c are real numbers", and not imaginary, or whatever.

Answer 6

€ means "is an element of the set"
N is the set of natural numbers {1, 2, 3,...}
Z is the set of integers {..., -2, -1, 0, 1, 2,...}
R is the set of real numbers

So in the problem you gave, it means a, b and c are real numbers.

Answer 7

enable a and b are the roots then sum of the roots a+b=-3/2..............i product ofn roots ab=-4/2=-2......................ii we choose new equation with a^2 and b^2 as roots (x-a^2)(x-b^2)=0 x^2 - (a^2+b^2)x + (ab)^2 =0..........................a million now sq. (i) a^2+b^2+2ab=9/4 a^2+b^2-4=9/4 a^2+b^2=25/4.............................. replace a^2+b^2 =25/4 and ab=-2 in 2 x^2-(25/4)x+4=0 4x^2 - 25x +sixteen=0 all the wonderful.