Question

Let a,b,c,d be non- zero integers. Show that each of the following is a rational number.
1. a/b + c/d
2. a/b - c/d
3. a/b * c/d
4. a/b / c/d

What do you conclude from these results?​

Answer 1

Given that;

a,b,c,d are non-zero integers

To prove that :

1. a/b + c/d
2. a/b - c/d
3. a/b * c/d
4. a/b / c/d      are rational numbers.

Proof: By the definition of rational number we say that,

" A number that can be represented in the form  $\frac{p}{q}$  of two integers such that q ≠ 0 is called a rational number."

Let us take a = 2/3 , b =3 , c=5 and d = 9. All these integers are rational numbers as they are in the form of $\frac{p}{q}$.

1. a/b + c/d = 2/9 + 5/9 = 7/9 is of the form p/q. So it is also a rational number.

2. a/b - c/d =  2/9 - 5/9 = -3/9 is of the form p/q. So it is also a rational number.

3. a/b * c/d = 2/9 * 5/9 = 10/81  is of the form p/q. So it is also a rational number.

4. a/b / c/d = 2/9 * 9/5 = 2/5  is of the form p/q. So it is also a rational number.

Thus, a/b + c/d , a/b - c/d , a/b * c/d and a/b / c/d are also a rational number.

so we can conclude that,

"For two rational numbers say x and y the results of addition, subtraction multiplication and division operations give a rational number."

Answer 2

Answer: We can conclude that the addition, subtraction, multiplication and division of rational numbers are always a rational number.

Given: a, b, c, d be non- zero integers.

To Find: Show that each of the following is a rational number.

Step-by-step explanation:

Step 1: Rational numbers are those numbers which are in the form of $\frac{p}{q}$ where $q\neq 0$. That means a number which has non-zero denominator must be a rational number.

Step 2: To verify numerically we can put different non-zero values in place of a, b, c and d to check our result. Let $a=-1$, $b=2$, $c=-7$ and $d=5$.

Step 3: So, $\frac{a}{b} =\frac{-1}{2}$ and $\frac{c}{d} =\frac{-7}{5}$

Putting the above values we get,

$1. a/b + c/d = (-1/2)+(-7/5)=-19/10\\ 2. a/b - c/d = (-1/2)-(-7/5)=9/10\\ 3. a/b * c/d =(-1/2)*(-7/5)=7/10\\ 4. a/b / c/d = (-1/2)/(-7/5) = 5/14$

Here we can see that all numbers are rational numbers.

So, we can conclude that the addition, subtraction, multiplication and division of rational numbers are always a rational number.

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