Thread: Equations

Hello,

for years i have accepted equations and knew how to use them. On the other day, i started thinking too much about equations....

if i have a quantity x and i know that x plus 2 equals 5, then i know that x equals 3. But why are equations always valid? How does saying that two things are equal, makes me able to extract the values that would satisfy them?

Thanks in advance

You can't say al the equations are valid, take for example:

This equation is not valid, why?

The purpose of an equation is that you're going to search a value of the variable wherefore the equation is true.

(Also I don't think this topic belongs in this section).

Equations *similar to* the example you gave are always true for rather advanced (yet fundamental) reasons.

Let's say you're working with the real numbers (R). They form a FIELD. I won't go into all the definitions and properties, but since R is a group under addition, you can always use "additive inverses" (i.e. subtraction) and preserve the equality.

Likewise, having multiplicative inverses allows us to divide.

That still does not explain why the statement of an equation allows me to find solutions =S

Originally Posted by DarkFalz

That still does not explain why the statement of an equation allows me to find solutions =S

Oh really? You sure seem to know a lot about this... for someone who is asking the questions.

Not all equations have solutions.
x + 1 = 0 has no solution in the naturals.
2x - 3 = 0 has no solution in the integers.
x^2 - 2 = 0 has no solution in the rationals.
x^2 + 2 = 0 has no solution in the reals.
x = x + 1 has no solution for almost every mathematical idea you've ever heard of for "x".

So what's the point of all this? We can constructively approach certain equations. In the case of a linear equation, we can solve for reasons in my first post. If you haven't taken the time to even search "ring/field/group", then it might not be best to call an explanation lacking...

Originally Posted by DarkFalz

Hello,

for years i have accepted equations and knew how to use them. On the other day, i started thinking too much about equations....

if i have a quantity x and i know that x plus 2 equals 5, then i know that x equals 3. But why are equations always valid? How does saying that two things are equal, makes me able to extract the values that would satisfy them?

Thanks in advance

the principle here was giving the name of "muqubalah" or "balancing" by the great algebraist al-Khwārizmī, or in plain english: equals to equals are equal.

x+2 = 5
x+2-2 = 5-2 (by the principle of balancing, stated above)
x+0 = 3 (calculating 2-2 and 5-3, elementary arithmetic operations)
x = 3 (the law of 0: adding nothing changes nothing).

now, if we look closely at what we just did, we find that the whole process depends on being able to subtract something from 2 that gives us 0. one way of saying this, is that -2 is an additive inverse for 2. so that is one important thing we need for "solving" equations, being able to "undo" an operation (in this case, addition).

the other important thing, is that we used a very special property of 0: x+0 = x, no matter which number we use for x. in this case, we say 0 is an identity for addition.

number systems need not possess inverses OR identities. for example, if we are limited to just positive (counting numbers), we CANNOT solve the equation:

x + 5 = 3

because 3 - 5 isn't any positive number.

another common operation used in equations is multiplication. so a similar problem might be:

3x = 5

here, we need a multiplicative inverse, to solve for x. if we are limited to just integers, there is no solution. but if we allow "fractions" (rational numbers), we have:

3x = 5
(1/3)(3x) = (1/3)5
(1/3)(3x/1) = (1/3)(5/1) (because anything divided by 1 is itself: 1 serves as a multiplicative identity, which means 1/1 = 1, since (1)(1) = 1).
(3x/3) = 5/3 (we multiply fractions by multiplying tops and bottoms)
(3/3)(x/1) = 5/3
1x = 5/3
x = 5/3

it is very possible to write equations in number systems without any solutions, for example:

x = x+1 which leads to:
0 = 1, which does not seem very reasonable.

in general, we say that a number system has to have a group structure, if we have only one operation, in order to solve equations. if we have two operations, we usually require what is called a ring structure (this is like a group-and-a-half, with some rules to make sure the two operations are compatible).

the best of all possible worlds, is what we call a field structure, where we have 2 group structures in one set of numbers (with some other rules that make fields "nice" to do algebra in). these important rules are:

for all a,b,c:

(a+b)+c = a+(b+c), called associativity of addition
a+b = b+a, called commutativity of addition
a+0 = 0+a = a, the identity law for 0
a+(-a) = -a+a = 0, the existence of inverses for addition

(ab)c = a(bc), the associative law of multiplication
ab = ba, the law of commutativity of multiplication
a1 = 1a = 1, the identity law of 1 for multiplication
a(1/a) = (1/a)a = 1, the existence of multiplicative inverses for all non-zero a (0 can't have an inverse, for special reasons)

a(b+c) = ab + ac, the law of distributivity of multiplication over addition (this is the "compatability" requirement).

these rules are the usual ones we need to have to start solving equations of most kinds. as you can see, they are very reasonable rules, but some things don't obey them. for example, functions don't always obey them. the "multiplication" in functions is called "composition", and functions don't always have "inverses". this means, in particular, that equations involving functions, can be very hard to solve, and sometimes impossible.