Apply the Closure property to predict the answer. Will the product of 0.6... and 7/8 be an irrational or rational number?
The product of 0.6... (a repeating decimal number) and 7/8 will result in a rational number. The Closure property of multiplication states that when any two rational numbers are multiplied, the result is always a rational number. Since 0.6... is a decimal representation of a rational number, and 7/8 is a rational number, their product will also be a rational number.
To apply the Closure property, we need to determine the type of numbers that are being multiplied.
First, let's consider the number 0.6... where the digit 6 repeats infinitely. This number can be written as 0.6666... or 2/3.
Next, we have the number 7/8, which is a fraction and can be written as a ratio of two integers.
When we multiply 0.6... (2/3) by 7/8, we are multiplying a rational number (2/3) by another rational number (7/8).
According to the Closure property, the product of two rational numbers is always a rational number.
Therefore, the product of 0.6... (2/3) and 7/8 will be a rational number.
To apply the Closure property in this case, we need to determine whether the product of 0.6... (repeating decimal) and 7/8 will result in a rational or irrational number.
Firstly, let's simplify the problem by expressing 0.6... as a fraction. Let's assume x = 0.6... and multiply it by 10 to eliminate the decimal part.
10x = 6.6...
Now, we subtract x from 10x to eliminate the repeating part.
10x - x = 6.6... - 0.6...
9x = 6
Solving for x:
x = 6/9
Reducing the fraction to its simplest form:
x = 2/3
So, 0.6... is equivalent to 2/3.
Next, we multiply 2/3 by 7/8:
(2/3) x (7/8) = (2 * 7) / (3 * 8) = 14/24
Reducing the fraction:
14/24 = 7/12
Since 7/12 can be expressed as a fraction, it is considered a rational number.
Therefore, applying the Closure property, the product of 0.6... and 7/8 is a rational number.