HELP. Help
I have to decide whether these are rational or irrational.
.73737373...
.61611611161111.....
1/23
Can someone explain this to me?
Rational numbers are just fractions. The decimal expansion of fractions will either terminate or it will become periodic after a while.
This means that the number:
.61611611161111.....
is not rational. The further you go in the decimal expansion, the more 1's will there be inbetween the sixes, so the digits will never repeat.
The two other numbers are, of course, rational.
Haha get noob
To determine whether a number is rational or irrational, we need to understand the concept of rational numbers.
Rational numbers can be expressed as fractions of two integers (a/b), where a and b are integers and b is not equal to zero. The decimal representation of rational numbers can either terminate or repeat after a certain point.
Let's examine the given numbers:
1. .73737373...
To check if this number is rational, we observe that the digits 73 repeat infinitely. Thus, we can express this number as a fraction. Let's use x to represent this number:
x = .73737373...
Multiplying both sides by 100 (to shift the decimal point two places to the right) gives:
100x = 73.73737373...
Now, subtracting the original equation (x) from this new equation (100x) cancels out the repeating part:
100x - x = 73.73737373... - .73737373...
This simplifies to:
99x = 73
Dividing both sides by 99:
x = 73/99
As 73/99 is a fraction, we can conclude that .73737373... is a rational number.
2. .61611611161111...
In this case, the digits do not repeat in a predictable pattern. As the decimal expansion continues, the number of 1's between the 6's increases. This means that there is no fractional expression that will result in this decimal representation. Therefore, .61611611161111... is an irrational number.
3. 1/23
Any fraction, such as this one, can be expressed as a rational number. In this case, 1/23 is already a fraction, so it is a rational number.
To summarize:
- The number .73737373... is rational because the digits repeat after a certain point.
- The number .61611611161111... is irrational because the digits never repeat in a predictable pattern.
- The fraction 1/23 is a rational number because it can be expressed as a ratio of two integers.