Show that each and every number is a rational number
A Rational Number can be made by dividing two integers.
(An integer is a number with no fractional part.)
1.5 is a rational number because 1.5=3/2
3.14 = 22/7. Most numbers we use are rational numbers: 5/1 7/3 1/2....
but surely not "each and every" number
There are many exceptions, such as π, √2, etc.
If by "number" you just mean integers, then yes, they are all rational numbers, with a denominator of 1.
To show that every number is a rational number, we need to prove that every number can be expressed as a ratio of two integers, where the denominator is not zero.
Let's start by considering whole numbers or integers. Any whole number is an integer and can be written as a ratio of itself and 1. For example, 5 can be written as 5/1, which is a ratio of two integers.
Next, let's consider fractions. A fraction is defined as the ratio of two integers. If we take any fraction, such as 3/4, we can see that it satisfies the definition of a rational number since it is expressed as the ratio of two integers.
Now, let's consider decimal numbers. Any decimal number can be written as a fraction by placing the decimal part over a power of 10. For example, the decimal number 0.25 can be written as 25/100, which simplifies to 1/4. This shows that decimal numbers can also be expressed as the ratio of two integers and are thus rational.
Lastly, let's consider irrational numbers such as π (pi) or √2 (square root of 2). These numbers cannot be expressed as the ratio of two integers. However, we can approximate them as decimal numbers, which can be expressed as the ratio of two integers. For example, π can be approximated as 3.14159, which is the rational number 314159/100000. So, even irrational numbers can be represented as ratios of integers, although they are not exact.
In conclusion, every number can be expressed as a rational number because it can be written as a ratio of two integers.