Rational vs irrational
Categorize Each example as rational or irrational
.66666666666666666...
.022022022022022022...
1.732...
4.589
16
5
.66666666666666666... - Rational (repeating decimal)
.022022022022022022... - Rational (repeating decimal)
1.732... - Irrational (√3)
4.589 - Rational (terminating decimal)
16 - Rational (whole number)
5 - Rational (whole number)
Place each number into the categories it belongs. (Some numbers will go into more than one category)
Natural
Whole
Integers
Rational
Irrational
Real
-13,0,81,3,1.2222...
Natural: 0, 3, 81
Whole: 0, 81
Integers: -13, 0, 3, 81
Rational: 0, 1.2222...
Irrational: None
Real: -13, 0, 81, 3, 1.2222...
Express 1/6 as a repeating decimal
To express 1/6 as a repeating decimal, we can use long division.
When we divide 1 by 6, we get a quotient of 0 and a remainder of 1. We then bring down a zero and divide 10 by 6, which gives us a quotient of 1 and a remainder of 4. We repeat this process until we identify a repeating pattern.
```
0.1
-----
6 | 1.000000...
- 0
-----
10
- 6
-----
40
- 36
-----
40
- 36
-----
40
- 36
-----
400
- 36
-----
400
- 36
-----
40
```
The division continues indefinitely, and we can see that the pattern "40" repeats without end. Therefore, 1/6 can be expressed as the repeating decimal 0.1666666... (The dots indicate that the digit 6 repeats indefinitely).
To categorize each example as rational or irrational, we need to understand the definitions of rational and irrational numbers.
1. Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They can be written as a terminating decimal or a repeating decimal.
2. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They cannot be written as a terminating or repeating decimal.
Now let's categorize each example:
1. .66666666666666666...
This is a repeating decimal, which means it can be expressed as a fraction. It can be written as 2/3. Therefore, it is rational.
2. .022022022022022022...
This is also a repeating decimal, which can be expressed as a fraction. It can be written as 2/90 or simplified as 1/45. Therefore, it is rational.
3. 1.732...
This is an example of the square root of 3, which is an irrational number. It cannot be expressed as a fraction or a repeating decimal. Therefore, it is irrational.
4. 4.589
This is a terminating decimal, which means it can be expressed as a fraction. It can be written as 4589/1000 or simplified as 229/50. Therefore, it is rational.
5. 16
This is an integer, and all integers are rational numbers because they can be expressed as a fraction by dividing them by 1. Therefore, it is rational.
6. 5
Similar to the previous example, this is also an integer. Therefore, it is rational.
To summarize:
.66666666666666666... is rational
.022022022022022022... is rational
1.732... is irrational
4.589 is rational
16 is rational
5 is rational
To categorize each example as rational or irrational, we need to understand the characteristics and definitions of these terms.
A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. In other words, a rational number can be written in the form of a/b, where a and b are integers.
An irrational number, on the other hand, is a number that cannot be expressed as a simple fraction. It cannot be written in the form of a/b, where a and b are integers.
Now let's categorize each example:
1. The number 0.66666666666666666... is a rational number. To see why, we can convert it to a fraction. Let x = 0.66666666666666666..., then 10x = 6.666666666666666..., and subtracting x from 10x, we get 10x - x = 6.66666666666666666... - 0.66666666666666666..., which simplifies to 9x = 6. By dividing both sides of the equation by 9, we get x = 6/9, which can be further simplified to x = 2/3. Since 2/3 is a fraction, the number 0.66666666666666666... is rational.
2. The number 0.022022022022022022... is also a rational number. Similar to the previous example, we can convert it to a fraction. Let x = 0.022022022022022022..., then 1000x = 22.022022022022022..., and subtracting x from 1000x, we get 1000x - x = 22.022022022022022... - 0.022022022022022022..., which simplifies to 999x = 22. By dividing both sides of the equation by 999, we get x = 22/999. Since 22/999 is a fraction, the number 0.022022022022022022... is rational.
3. The number 1.732... is an irrational number. This decimal represents the square root of 3 (√3). The square root of 3 is not a rational number, which means it cannot be expressed as a fraction. Therefore, 1.732... is irrational.
4. The number 4.589 is a rational number. It can be expressed as the fraction 4589/1000.
5. The number 16 is a rational number. It can be expressed as the fraction 16/1.
6. The number 5 is a rational number. It can be expressed as the fraction 5/1.
In summary:
- 0.66666666666666666... is a rational number.
- 0.022022022022022022... is a rational number.
- 1.732... is an irrational number.
- 4.589 is a rational number.
- 16 is a rational number.
- 5 is a rational number.