Question

Approximate irrational number 34 to the nearest 0.05

Answer 1

First of all, 34 is a rational number: $\frac{34}{1}$
If you mean the square root of 34 (and I think you do), then here is one way to approximate it:
We know that the √25 is 5 and the √6 = 36 and that the √34 is somewhere between those two numbers.  It is certainly closer to 36 than 25 so I am going to guess high. I will guess 5.8  I will do 34 ÷ 5.8 = (about) 5.8620.  That was close but not rounded to the nearest .05 since it could be 5.80 (my guess) or 5.90 (5.86 rounded to the nearest 05.
Let's do it one more time:   I will find the average of the the number I divided by and the quotient:  (5.8 + 5.6820) ÷ 2 = 5.831  I will use this as my new divisor with 34
34 ÷ 5.831 = (about 5.8309)
You can see that I now know the √34 out to the nearest hundred.  It is about 5.83
Notice that as many times as I could keep doing this procedure, I will never get a number that, when multiplied by itself, gives me exactly 34.  That is the definition of an irrational number.