sum or product of two rational numbers is
rational
also a rational number.
To see why this is true, let’s first define rational numbers. A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. For example, 2/3, -4/5, and 7/1 are all rational numbers.
Now let’s consider the sum of two rational numbers. Let’s say we have two rational numbers a/b and c/d, where a, b, c, and d are integers and b and d are not zero. To find their sum, we can use a common denominator by multiplying both fractions by bd:
a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd
Since the numerator and denominator of the resulting fraction are both integers (the sum of two integers is always an integer), we can conclude that the sum of two rational numbers is also a rational number.
Similarly, let’s consider the product of two rational numbers. Again, let’s say we have two rational numbers a/b and c/d. To find their product, we simply multiply the numerators and denominators:
(a/b) x (c/d) = ac/bd
Again, since the numerator and denominator of the resulting fraction are both integers (the product of two integers is always an integer), we can conclude that the product of two rational numbers is also a rational number.
Therefore, we can say that the sum and product of two rational numbers is always a rational number.
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