Keyshia said that one of the following numbers is IRRATIONAL. Which number is IRRATIONAL and why is it IRRATIONAL?
(-2/3,2/3,0.666...,or 0.676676667...)
To determine which number is irrational among the given options, we need to understand what an irrational number is.
An irrational number is a real number that cannot be expressed as a simple fraction or a ratio of two integers. It cannot be written as terminating or repeating decimals.
Now let's analyze each option to identify the irrational number:
1. -2/3: This is a fraction, and therefore it is a rational number. It can be expressed as a ratio of two integers (-2 and 3).
2. 2/3: Similar to the previous option, this is a fraction and is considered a rational number. It can also be expressed as a ratio of two integers (2 and 3).
3. 0.666...: This decimal represents a repeating pattern of 6's. Although it may seem to go on forever, it is still a rational number. It can be expressed as the fraction 2/3 (since 6/9 simplifies to 2/3).
4. 0.676676667...: This decimal appears to have a repeating pattern as well, but it is not a simple repeating pattern like option 3. To determine if it is rational or irrational, we need to check if it repeats or terminates.
To do this, we can convert this decimal into a fraction. Let's assign the decimal value as 'x' and subtract it from 10x.
10x - x = 10x - 0.676676667...
This simplifies to:
9x = 6.999...
Now let's subtract these equations:
9x - x = 6.999... - 0.676676667...
This simplifies to:
8x = 6.322323333...
Now we divide both sides by 8:
x = 0.790290416...
As we can see, x is a non-repeating decimal that does not terminate, meaning it cannot be expressed as a simple fraction or a ratio of two integers. Therefore, it is an irrational number.
Based on our analysis, the number 0.676676667... is the irrational number among the given options.
Note: The process explained above is a way to check if a decimal is rational or not by converting it into a fraction. In this case, when the decimal didn't convert into a fraction, it is concluded that the number is irrational.