Let 𝑎 and 𝑏 be rational numbers. Show that 13 𝑎 + 23 𝑏 is a rationalnumber.

Sure, here is the step by step proof:

1. By definition, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since a and b are given to be rational numbers, we can express them as a = p1/q1 and b = p2/q2 where p1, q1, p2, q2 are integers and q1, q2 ≠ 0.

2. The operation of addition and multiplication is closed under integers. This means that the sum or product of any two integers will always be an integer.

3. Therefore, 13a + 23b can be rewritten as 13(p1/q1) + 23(p2/q2) = (13p1/q1) + (23p2/q2) = (13p1q2 + 23p2q1) / (q1q2).

4. The numerator (13p1q2 + 23p2q1) and the denominator (q1q2) are both integers because they are the sum and product of integers respectively.

5. Since the quotient of two integers with non-zero denominator is a rational number, 13a + 23b is a rational number.