A repeating decimal like 0.2222222.... is a Response area number and can be written as the fraction Response area.
A stopping decimal like 0.22 is a rational number and can be written as the fraction Response area.
A non-stopping and non-repeating decimal like 0.1234567891011... or 0.121121112... is a Response area number can be written as the fraction Response area.
The square root of a Response area is a rational number and also a whole number.
The square root of a Response area is an irrational number.
A repeating decimal like 0.2222222... can be written as the fraction Response area by following a simple process. Let's use x to represent the repeating decimal.
Step 1: Multiply the repeating part x by 10^n, where n is the number of digits in the repeating part.
10^n * x = 0.2222222... * 10^n
Step 2: Subtract the original number x from the product obtained in step 1 to eliminate the repeating part.
10^n * x - x = 0.2222222... * 10^n - x
Simplifying the equation:
10^n * x - x = 2.2222222... - 0.2222222...
On the right side, subtracting the decimal part from the repeating part leaves:
10^n * x - x = 2
Now, factor out x on the left side:
x * (10^n - 1) = 2
To isolate x, divide both sides by (10^n - 1):
x = 2 / (10^n - 1)
So, the fraction representation of the repeating decimal 0.2222222... is 2 / (10^n - 1), where n is the number of digits in the repeating part.
Similarly, for a stopping decimal like 0.22, we can write it as a rational number by following a similar process.
Step 1: Let x represent the stopping decimal.
Multiply it by a power of 10 that eliminates the fractional part, which in this case is 100 (2 decimal places).
100 * x = 0.22 * 100
Step 2: Simplify the equation:
100 * x = 22
Now, divide both sides by 100 to isolate x:
x = 22 / 100
Simplifying the fraction:
x = 11 / 50
Therefore, the fraction representation of the stopping decimal 0.22 is 11 / 50.
Moving on to non-stopping and non-repeating decimals like 0.1234567891011... or 0.121121112..., they are considered irrational numbers. This means they cannot be expressed as fractions with integer numerator and denominator.
Next, the square root of a rational number and whole number, let's say x, can be written as a rational number if the square root of x is also a whole number.
For example, if x is a perfect square like 4, then the square root of x is 2, which is a whole number and hence a rational number.
On the other hand, irrational numbers have square roots that are non-repeating, non-stopping decimals. They cannot be expressed as fractions and the decimal representation continues infinitely without a pattern.
For instance, the square root of 2 is an irrational number and its decimal representation is approximately 1.41421356... This number cannot be expressed exactly as a fraction.
In summary, a repeating decimal can be written as a fraction by following a specific process, while a stopping decimal can be directly converted to a fraction. Non-stopping and non-repeating decimals are considered irrational numbers. Square roots can be rational or irrational, depending on whether they result in whole numbers or non-repeating decimals.