Question

explain why the sum of two rational numbers is closed under addition. give an example to make your case.

Explanation:

Define rational numbers

Using the Rational Numbers knowledge point, any rational number can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b
eq 0\). Let our two rational numbers be:

$$x = \frac{a}{b} \quad \text{and} \quad y = \frac{c}{d}$$

where \(a, b, c, d\) are integers, and \(b, d
eq 0\).

Add the two numbers

Using the Closure Under Addition knowledge point, we find the sum of \(x\) and \(y\) by finding a common denominator:

$$x + y = \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Analyze the resulting sum

Since integers are closed under multiplication and addition, the numerator \(ad + bc\) is an integer. Similarly, the denominator \(bd\) is a non-zero integer because \(b
eq 0\) and \(d
eq 0\). Since the sum \(\frac{ad + bc}{bd}\) is a ratio of two integers with a non-zero denominator, it is a rational number.

Provide a concrete example

Let \(x = \frac{1}{2}\) and \(y = \frac{2}{3}\), which are both rational numbers.

$$x + y = \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}$$

Since \(7\) and \(6\) are integers and \(6
eq 0\), the sum \(\frac{7}{6}\) is also a rational number.

Answer:

A rational number is defined as any number that can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b
eq 0\).

When adding two rational numbers, \(\frac{a}{b}\) and \(\frac{c}{d}\):

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Because integers are closed under addition and multiplication, both the numerator (\(ad + bc\)) and the denominator (\(bd\)) must be integers. Since \(b
eq 0\) and \(d
eq 0\), the product \(bd
eq 0\). Therefore, the sum is a ratio of two integers with a non-zero denominator, which is by definition a rational number.

Example:

$$\frac{1}{2} + \frac{2}{3} = \frac{3 + 4}{6} = \frac{7}{6}$$

Since \(\frac{1}{2}\) and \(\frac{2}{3}\) are rational, and their sum \(\frac{7}{6}\) is also a rational number, this demonstrates closure under addition.