sum or product of two rational numbers is
rational
also a rational number.
To understand why, we need to define what rational numbers are. A rational number is any number that can be written as a ratio of two integers, where the denominator is not zero.
Now let’s consider the sum of two rational numbers. Let’s say we have two rational numbers, a/b and c/d. To find their sum, we need to add their numerators and keep the same denominator:
(a/b) + (c/d) = (ad + bc) / bd
The numerator (ad + bc) is also an integer, since it is the sum of two integers. The denominator (bd) is also an integer, as long as neither b nor d is zero. Therefore, 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 need to multiply their numerators and denominators:
(a/b) x (c/d) = (ac) / (bd)
Again, both the numerator (ac) and the denominator (bd) are integers, as long as neither b nor d is zero. Therefore, the product of two rational numbers is also a rational number.
In conclusion, the sum or product of two rational numbers is always a rational number, since it can be expressed as a ratio of two integers.
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