show/hide this revision's text 2 New answer as I was way-off.

My approach would be

Thanks to ask them Pete for the link on what it would mean to solve various equationsthe question means. I am not sure what age we was misled by the other responses.

Here are talking about or what background knowledge they havetwo thoughts:

i.

The equations could include algebraic equations but more challenging would be to ask them what a number like $5^{\sqrt 2}$ What is the square root of 14? Well, it's between 3 and 4. They Then we can probably type this into their calculators. You could ask them to solve $\sin(x)= 1-x^2$ (off do better by taking the top average of my head) or possibly $\exp(\exp(x))-\exp(x)+1=0$. The idea is to produce equations which obviously have solutions because you 3 and 14/3. Then we can plot graphs repeat this and see lines crossing but where it is also obvious that there is not going to be a formula for the solutiondo even better.

This raises

ii. Take the problem of reconciling these two points Fibonnaci sequence 1,1,2,3,5,8,13,... Take ratios of viewsuccessive terms 1/1,1/2,2/3,3/5,5/8,8/13,... Then you resolve this by showing that you can find then these are approximations to the solution and the approximations can be made better "golden ratio".

A personal anecdote: My daughter is 11 yrs old, bright and betterinterested in maths.

You could also discuss Archimedes approach to calculating the area of I tried i. on her and drew a circleblank look.

The point is I conclude from this that you can work out the decimal expansion of "the number" either it is a mistake to as many decimal places as you wantteach your own children or that she was not ready for this.
show/hide this revision's text 1

My approach would be to ask them what it would mean to solve various equations. I am not sure what age we are talking about or what background knowledge they have.

The equations could include algebraic equations but more challenging would be to ask them what a number like $5^{\sqrt 2}$ is. They can probably type this into their calculators. You could ask them to solve $\sin(x)= 1-x^2$ (off the top of my head) or possibly $\exp(\exp(x))-\exp(x)+1=0$. The idea is to produce equations which obviously have solutions because you can plot graphs and see lines crossing but where it is also obvious that there is not going to be a formula for the solution.

This raises the problem of reconciling these two points of view. Then you resolve this by showing that you can find approximations to the solution and the approximations can be made better and better.

You could also discuss Archimedes approach to calculating the area of a circle.

The point is that you can work out the decimal expansion of "the number" to as many decimal places as you want.