Non-terminating Decimal Numbers
1. 1/3 = 0.3333...
When we divide 1 by 3, we get 0.3333... as the decimal expansion. This decimal expansion is non-terminating because it repeats the digit 3 infinitely. This is because 3 does not divide evenly into 1, leading to a repeating decimal pattern.
2. 1/7 = 0.142857142857...
Similarly, when we divide 1 by 7, we get 0.142857142857... as the decimal expansion. This decimal expansion is also non-terminating but repeating. The pattern 142857 repeats infinitely in this case.
3. √2 = 1.414213562373095...
The square root of 2, represented as √2, has a decimal expansion that is non-terminating and non-repeating. This means that the digits after the decimal point continue indefinitely without following a specific repeating pattern. The decimal expansion of √2 is an example of an irrational number, which cannot be expressed as a simple fraction.
In conclusion, these three examples illustrate different types of non-terminating decimal numbers. The first two examples have repeating patterns, while the third example is non-repeating. These types of decimal expansions are common in mathematics and highlight the diversity of numbers beyond simple integers or fractions.