AI Tutor
If you add two simple fractions together, could the answer ever spit out an endless, non-repeating number like Ο€\pi or 2\sqrt{2}? It turns out the fraction world has an unbreakable barrier, and we can prove it with basic algebra.
A rational number is any number that can be written as a fraction ab\frac{a}{b}, where aa and bb are integers (whole numbers like βˆ’3,0,7-3, 0, 7) and bβ‰ 0b \neq 0.
What happens when we take two completely unknown rational numbers and add them together?

The Algebraic Proof

Let our first rational number be x=abx = \frac{a}{b} and our second be y=cdy = \frac{c}{d}, where a,b,c,a, b, c, and dd are integers with b≠0b \neq 0 and d≠0d \neq 0.
To add these fractions, we find a common denominator by multiplying the top and bottom of each fraction by the other's denominator: aβ‹…dbβ‹…d+cβ‹…bdβ‹…b=ad+bcbd\frac{a \cdot d}{b \cdot d} + \frac{c \cdot b}{d \cdot b} = \frac{ad + bc}{bd}.
Because multiplying and adding integers always results in another integer, the top expression ad+bcad + bc is definitely an integer.
Since neither bb nor dd is zero, their product bdbd is also a non-zero integer, meaning ad+bcbd\frac{ad + bc}{bd} meets the exact definition of a rational number.
Does this algebraic blueprint match what happens when we use actual numbers?

Numerical Walkthrough

Let's test x=23x = \frac{2}{3} and y=45y = \frac{4}{5}, where a=2,b=3,c=4,a = 2, b = 3, c = 4, and d=5d = 5.