What Rationalizing the Numerator Does
Rationalizing the numerator rewrites a fraction so that no square root remains on top. It is the mirror image of rationalizing the denominator, and it is the standard trick for simplifying limits in calculus, where a radical difference like \(\sqrt{x+h} - \sqrt x\) in the numerator otherwise creates a stubborn 0/0 form. This calculator handles two shapes: a single radical numerator \(\sqrt a / b\) and a binomial (conjugate) numerator \((c + \sqrt d) / b\).
How to Use It
Pick the numerator type. For the simple case enter the value a under the square root and the denominator b. For the conjugate case enter the rational part c, the value d under the root, and the denominator b. The tool returns the rationalized (radical-free) numerator and the exact decimal value of the whole expression, which never changes because you are multiplying by a form of 1.
The Formula Explained
For a single radical, multiply the top and bottom by \(\sqrt a\). Because \(\sqrt a \cdot \sqrt a = a\), the numerator becomes the rational number a:
$$\frac{\sqrt a}{b} = \frac{a}{b\,\sqrt a}$$For a binomial numerator, multiply by the conjugate \(c - \sqrt d\). The difference-of-squares pattern \((c + \sqrt d)(c - \sqrt d) = c^2 - d\) clears the radical, so the numerator becomes the rational number \(c^2 - d\):
$$\frac{c + \sqrt d}{b} = \frac{c^2 - d}{b\left(c - \sqrt d\right)}$$Worked Example
Take \((2 + \sqrt 3) / 5\). The conjugate of the numerator is \(2 - \sqrt 3\), and the new numerator is \(2^2 - 3 = 1\). So the expression equals
$$\frac{2 + \sqrt 3}{5} = \frac{1}{5\left(2 - \sqrt 3\right)} \approx 0.7464.$$The radical has moved out of the numerator, and the decimal value is unchanged. The calculator reports the rationalized numerator 1 and the value 0.7464.
Frequently Asked Questions
Why would I rationalize the numerator? It is the key step for evaluating limits such as \(\lim_{h \to 0} (\sqrt{x+h} - \sqrt x)/h\); clearing the radical on top cancels the h and removes the 0/0 form.
Does rationalizing change the value? No. You multiply by the conjugate over itself, which equals 1, so the number stays the same and only its written form changes.
What if c squared minus d is negative? That is fine. The rationalized numerator is still a rational number; a negative sign simply carries through, and the decimal value stays correct.
