Why does $\frac {1}{a}-\frac{1}{b}=\frac {b-a}{ab}$? I really don't understand the expression.
$$\frac {1}{a}-\frac{1}{b}=\frac {b-a}{ab}$$
I generally have a hard time understanding non-intuitive things in math and this is one of them. Normally when I don't understand something I use an app , photomath, to help explain expressions/equations I don't understand however, I still need help with this expression.
I'm told that to get to $\frac {b-a}{ab}$ you need to expand the fraction to the least common denominator:
$$\frac {1}{a}-\frac{1}{b} \to \frac {\pmb b\times 1}{\pmb b a} - \frac {\pmb a \times 1}{\pmb a b} \to \frac {b}{ab} - \frac {a}{ab} \to \frac {b-a}{ab}$$
What I don't understand is this
$$ \frac {\pmb b \times 1}{\pmb b a} - \frac{\pmb a \times 1}{\pmb a b}$$
I don't understand how exactly the $a$ and $b$ seemingly 'appear' in the expression.
It is easier to grasp if you start with the RHS.
Note that $$\frac {b-a}{ab} = \frac {b}{ab} -\frac {a}{ab}=$$
$$ \frac {1}{a} -\frac {1}{b}$$
Original Equation 
$\frac{1}{a} - \frac{1}{b} \stackrel{?}{=} \frac{b - a}{ab}$
Multiply both sides by $ab$ 
$ab \cdot \big(\frac{1}{a} - \frac{1}{b}\big) \stackrel{?}{=} ab \cdot \big(\frac{b - a}{ab}\big)$
Distribute the $ab$ 
$\frac{ab}{a} - \frac{ab}{b} \stackrel{?}{=} \frac{ab \cdot (b - a)}{ab}$
Cancel out variables and both sides have identical expressions 
$b - a \stackrel{\checkmark}{=} b - a$