So, what are rational numbers?
Mathematicians define rational numbers as any number that can be written in the form of a ratio of two integers. Explained even better, any number is rational if you can write it as a fraction, where both the numerator and denominator are integers. A number is considered an integer if it’s a whole number (both negative and positive and even zero). Some integer examples include: 45563,34; 2; 0; -345; but also: 2/3; 4.000005, 3.7, PI and -9.09 are not. In simple words, an integer is in the set of {…3,-2,-1,0,1,2,3,…}, where the dots are translated into numbers that can go in the negative and positive direction as well
If you are still wondering why they are called rational numbers, the term “rational” comes from “ratio” because these numbers are the ones that can be written in a ratio form (a/b) where a and b are integers. The term irrational names all the numbers that are not rational. So, basically, every integer is a rational number due to the fact that every integer can be written by using “/1”. Let’s see some examples:
- 7 is a rational number because it can be written as 5/1
- 1/3 is a rational number because they are fractions with both the numerator and denominator as integers
- 3434324/664543 is a rational number because they are fractions with both the numerator and denominator as integers
- -4/7 is a rational number because they are fractions with both the numerator and denominator integers, even if the number is a negative one.
But how can numbers with decimals be rational numbers?
In order to best explain how a number like 3.75 is a rational number, you should follow the next example carefully. So, basically the number 3.75 can be written as 375/100, right? Good, so this makes the decimal number into a fraction with both the numerator and denominator being integers. Or you can write the number as 750/200 or even 3 and 3/4s and even 15/4. These are just some variations to understated that this is clearly a rational number because again, it can be written in two integers.
The number line and rational numbers
Measuring is the most important aspect about rational numbers. In other words, naming any distance from 0 along the number line is basically such a number. Mathematicians asked themselves if rational numbers can account for EVERY distance from 0? In order to pursuit this particular question they used the following theorem: any two rational numbers have the same ratio as natural numbers. This has proven to be true because fractions having the same denominator possess the same ratio as their numerators. This means that we can always express two fractions with the same denominator (2/5 : 3/5 = 2:3, which means that 2/5 is two-thirds of 3/5)
This example and many others prove that any two rational numbers have the same ratio as two natural numbers.
Why are rational numbers so important?
Mathematicians in ancient history discovered that it is always possible to find another rational number right between any two given members in a set of rational numbers. This means that rational numbers are a continuous set, but the numbers are countable too. From this premises, according to Honsberger’s book in 1991, we can establish that for any given different rational numbers a,b,c we can say that the 1/((a-b)^2)+1/((b-c)^2)+1/((c-a)^2) can be considered the square of the rational number (a^2+b^2+c^2-ab-bc-ca)/((a-b)(b-c)(c-a)). Even if it might sound complicated, try to use various rational numbers instead of the a,b,c letters and see that the formula is actually true. This can help learners of math to carry out various exercises in order to practice their skills.
The ancient Greek mathematicians thought that all the things in the world can be measured just by using rational numbers. Sadly, the Pythagorean Theorem came and proved that some lengths cannot be written as a simple rational number so they had to change their perspectives. However, rational numbers are used in a great deal of ways: from buying the groceries at the store around the corner to selling merchendise in a vinyl shop and many other activities like these ones. This is because it implies buying and selling things with money.
There must be irrational numbers as well!
Totally opposite from the explanation given for rational numbers, the irrational ones cannot be expressed as a fraction (a/b) for any integers a and b. Irrational numbers are considered to have decimal expansions that cannot terminate or become periodic. Basically, every transcendental number is an irrational number (a number that is not the root of any integer polynomial, which means that it is not an algebraic number of any sort).
The most famous irrational number is sqrt(2), known by many as Pythagoras’s constant. The lure talks about Hippasus, a Pythagorean mathematician and philosopher that used various geometric methods in order to demonstrate that sqrt(2) is an irrational number. This all happened while he was at sea and after all the crew members (who were Pythagoreans) found out about it, they threw him over board. Other examples include e, pi, the Golden Ratio and so on. As a conclusion here, irrational numbers have endless digits that are not repeating themselves to the right of the decimal point.
Given the highlights discussed above, you can say that there is always an irrational number between any two given rational numbers. This proves that the irrational numbers are not just a special case such as pi or e, but they are present in the mathematical universe as an infinite number as well. Oh, and one thing to keep in mind here: if you are trying to add an irrational number with a rational one, the result will always be an irrational number.
Multiplying irrational numbers
One note on the idea of multiplying irrational numbers is the fact that the result will not always be an irrational number. Let’s see some examples: π × π = π2 is an irrational number for sure, but if you multiply √2 × √2 = 2 which is a rational number. So, as you can see, the presumption that by multiplying irrational numbers you will get a rational one is not quite true.
Conclusion on rational and irrational numbers
As a conclusion to this segment of rational numbers, in order to further understand both rational and irrational numbers, you should really know the main difference: the first case are the rational ones which can be written in form of a fraction with both the nominator and denominator as integers; whereas the irrational numbers are expressed in an infinite number of digits. In order to get a better grasp on how the numbers work, you should take up some spotting exercises to determine which numbers are rational and which aren’t. In this way, you can get a better grasp of the concept and understand how it works from your examples.
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