Proof that 4/3 is a Rational Number

Rational Numbers
- Rational numbers are those numbers that can be expressed in the form of p/q where p and q are integers and q is not equal to zero.
- Rational numbers can also be expressed as terminating decimals or repeating decimals.

Proof that 4/3 is a Rational Number
- To prove that 4/3 is a rational number, we need to show that it can be expressed in the form of p/q where p and q are integers and q is not equal to zero.
- Let us assume that 4/3 is not a rational number.
- This means that 4/3 cannot be expressed as the ratio of two integers.
- Let us express 4/3 as a decimal.
- Dividing 4 by 3, we get 1.33333…
- The decimal representation of 4/3 is a repeating decimal.
- Any repeating decimal can be expressed as a rational number.
- Let x = 1.33333…
- Multiplying both sides by 10, we get 10x = 13.33333…
- Subtracting x from 10x, we get 9x = 12
- Dividing both sides by 9, we get x = 4/3
- Therefore, 4/3 can be expressed as the ratio of two integers.
- Hence, 4/3 is a rational number.

Irrational Numbers
- Irrational numbers are those numbers that cannot be expressed as the ratio of two integers.
- Irrational numbers cannot be expressed as terminating decimals or repeating decimals.
- Examples of irrational numbers are √2, √3, π, e, etc.

In conclusion, 4/3 is a rational number and can be expressed as the ratio of two integers. It is not an irrational number.