Question

based on this sum and using the closure property of integers what conclusion can you make about the sum of two rational numbers. explain your answer

Answer 1

Answer:

The closure property of the integers means that if we have a and b, integers then:

a + b = c will also be an integer.

(i will also use the fact that if a and b are integers, then  a*b is also an integer)

Now suppose that we have two rational numbers.

a/b and c/d.

where a, b, c and d are integers. (b and d are different than zero)

the sum of those two numbers can be written as:

$\frac{a}{b} + \frac{c}{d} = \frac{a*d + b*c}{b*d}$

Now, a*d, b*c and b*d are integers (because integers are closed under multiplication)

and because a*d and b*c are integers, then:

a*d + b*c is also an integer.

Then:

(a*d + b*c)/(b*d)

is the quotient between two integers, which is a rational number.

Then we can conclude that the rational numbers are closed under the addition operation.