How does “rationalizing the denominator” use the form of 1?

"Rationalizing the denominator" of a number is exactly what its title implies: turning its denominator into a rational number.

For an example, let's take the fraction 1/√2. We call √2 irrational because it cannot be expressed as a ratio of two integers. Oftentimes when we have irrational numbers like √2 in the denominator, it helps to perform a little bit of algebraic manipulation on them to make the denominator rational. Here's where that "form of 1" comes in. Remember that, as a rational number, 1 can be expressed as n/n, where n is any number. We can use this fact to rationalize the denominator of 1/√2 by multiplying it by √2/√2, which is equivalent to 1.

When we multiply the two together, we get:



Crucially, multiplying by 1 in the form of √2/√2 keeps the fraction's value the same, since 1 times any number just equals that same number. If we weren't multiplying but some form of 1, we'd change the number entirely, making the whole attempt at rationalizing the denominator pointless.