To understand how √2 equals approximately 1.4142, we can use various mathematical techniques. Here's a broad explanation:

1. Geometric Method:
Imagine a right-angled triangle with both legs equal to 1 unit. Using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs, and c is the hypotenuse.

1^2 + 1^2 = c^2
2 = c^2
c = √2

Now, imagine a square with a side length of 1 unit. The diagonal of this square is equal to √2 units.

Using a compass and straightedge, we can construct a sequence of triangles and squares to approximate the value of √2.

2. Decimal Expansion:
We can express √2 as a decimal using the following method:

√2 = 1 + 0.4 + 0.02 + 0.0004 + ... (using the Babylonian method or other numerical methods)

This decimal expansion converges to approximately 1.4142.

3. Continued Fractions:
We can express √2 as a continued fraction:

√2 = 1 + 1/(2 + 1/(2 + 1/(2 + ...)))

This continued fraction converges to approximately 1.4142.

4. Babylonian Method:
This ancient method involves making an initial guess, then iteratively refining that guess using the formula:

x(n+1) = (x(n) + 2/x(n)) / 2

Starting with an initial guess of 1, we can iterate this formula to converge to approximately 1.4142.

These mathematical techniques demonstrate how √2 equals approximately 1.4142.