Let a/b and c/d represent two rational numbers. This means a, b, c, and d are
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, and
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. The product of the numbers is acbd, where bd is not 0. Because integers are closed under
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, acbd is the ratio of two integers, making it a rational number.
a) whole numbers
b) non-zero whole numbers
c) multiplication
Let's break down the steps to understand the given information:
Step 1: Rational Numbers
Let a/b and c/d represent two rational numbers. This means that a, b, c, and d are integers, and b and d are not equal to 0.
Step 2: Product of the Numbers
The product of the given rational numbers is acbd.
Step 3: Closure Property of Integers
The closure property states that when two integers are multiplied, the result is also an integer.
Step 4: acbd is Rational
Since acbd is the product of two integers, it follows the closure property. Therefore, acbd is the ratio of two integers, making it a rational number.
In summary, when a/b and c/d are rational numbers, the product of the numbers (acbd) is also a rational number due to the closure property of integers.
To understand and verify why the product of two rational numbers is also a rational number, let's break down the given information and explain the concepts step by step.
1. Rational numbers: Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. In this case, the two rational numbers are represented as a/b and c/d, where a, b, c, and d are integers and b and d are not zero.
2. Product of rational numbers: The product of two rational numbers is obtained by multiplying their numerators and denominators. Mathematically, the product of a/b and c/d is (a * c) / (b * d).
3. Closure under multiplication: It is important to note that integers are closed under multiplication, which means that the product of any two integers is also an integer. By multiplying the integers a and c as well as b and d, we will obtain two integers ac and bd.
4. Ratio of two integers: Now, let's consider the product acbd. Since ac and bd are both integers (as stated in point 3), their product is also an integer. Therefore, acbd is the product of two integers, making it an integer.
5. Rational number: Since the product of the two rational numbers a/b and c/d, represented by acbd, is an integer (as explained in point 4), and integers are closed under multiplication (as stated in point 3), acbd also satisfies the criteria for a rational number because it can be expressed as the ratio of two integers.
In conclusion, the product of two rational numbers, a/b and c/d, is obtained by multiplying their numerators and denominators, resulting in the product ac/bd. Since integers are closed under multiplication, the product of the integer fractions ac and bd is also an integer. Hence, acbd is the ratio of two integers, making it a rational number.