Math Is Fun Forum

@anonimnystefy: Nope.

@Agnishom: Nope.

Try again.

Nope. You were actually closer the first time.

debjit625, I'm sorry for my bad-tempered post above. I suppose I had had a bad day and somehow let it spill over onto the forum. It was totally wrong of me and I apologize.

Anyway, "condition" simply means something that has to be fulfilled in order for a given statement to be true. For example, suppose the question was: "What's the condition under which the quadratic equation

,

, has no real roots?" Then the condition would be:

. In order for the statement "

,

, has no real roots" to be true, the condition

has to be fulfilled.

In this particular question, the statement is "

(

) has roots in the ratio

". In order for this to be true, we must have

. This is the condition that has to be fulfilled in order for the given statement to be true.

Okay, suppose the car is travelling with velocity

, the tree has diameter

and is at distance

from the road; suppose the hidden person is hugging the tree, and that, looking down from above, the person, car and centre of the cross section of the tree trunk are always in a straight line.

Then I make it the speed of the person when the distance between the tree and the car is

is

.

Let the roots be

and

. Thus

Eliminating

gives

as the required condition.

Test:

have roots in the ratio

.

Oops, pardon me.

No,

is constructed by means of either Dedekind cuts or equivalence classes of Cauchy sequences of rationals. The process is very different and much more complicated because you are now constructing an uncountable set from a countable one.

It depends on how you are defining your sets of numbers. If you are defining them by axioms, then there is nothing to prove since the axioms will include commutativity and associativity of addition. But if you are constructing them from smaller sets of numbers, then commutativity and associativity need to be proved.

For example, here is briefly how

is constructed from

. Define a relation

on

by

iff

. (Hint: Think of

as the "difference"

.) Then

is an equivalence relation and

is defined as the set of all equivalence classes under

. Let the equivalence class containing

be denoted

. Then addition in

is defined as

After checking that the operation is well defined, one can proceed to verify that

is commutative and associative.

Moreover, zero

in

is the equivalence class

and multiplication in

is defined as

.

Similarly

can be constructed from

as equivalence classes of the equivalence relation

on

defined by

iff

. Letting the equivalence class containing

be denoted

addition in

is definied as

The sequence is absolutely convergent iff both and converge.

It is conditionally convergent iff converges while diverges.

Examples.

is absolutely convergent. We have and .

is conditionally convergent. We have while is divergent.

So where are the imaginary numbers?

Hi Al-Allo. Your equations are wrong. They should be as follows