Understanding the Problem
To find the number (let's call it \( x \)) where the difference between \( x \) and its positive square root \( \sqrt{x} \) is 12, we set up the following equation:
\[ x - \sqrt{x} = 12 \]

Rearranging the Equation
We can rearrange this equation to isolate the square root:
\[ x - 12 = \sqrt{x} \]
Now, we will square both sides to eliminate the square root:
\[ (x - 12)^2 = x \]

Expanding the Equation
Now expand the left side:
\[ x^2 - 24x + 144 = x \]

Simplifying the Equation
Next, we simplify the equation by moving all terms to one side:
\[ x^2 - 25x + 144 = 0 \]

Using the Quadratic Formula
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = 1 \)
- \( b = -25 \)
- \( c = 144 \)
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values:
\[ x = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 1 \cdot 144}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 625 - 576 = 49 \]
Now substituting back:
\[ x = \frac{25 \pm 7}{2} \]

Finding the Solutions
Calculating the two potential solutions:
- \( x = \frac{32}{2} = 16 \)
- \( x = \frac{18}{2} = 9 \)

Verifying the Solutions
- For \( x = 16 \): \( 16 - \sqrt{16} = 16 - 4 = 12 \) (Valid)
- For \( x = 9 \): \( 9 - \sqrt{9} = 9 - 3 = 6 \) (Invalid)

Conclusion
Thus, the number is:

16