A common misconception is that 0 is not a rational number. It’s not difficult to see why this idea is so popular: the decimal representation of 0 does not have any repeating digits and it doesn’t end in an infinite sequence of 9s. However, this idea only appears true when you examine the decimal representation of 0 without considering other representations such as binary, octal or hexadecimal. In order to understand if zero is a rational number we must consider all its representations simultaneously.
This blog post will focus on whether or not zero can be considered a rational numbers by discussing how each representation affects our understanding of the question at hand.
The word “rational” is defined by the dictionary as “sensible and practical, not extreme or excessive.” Therefore to be a rational number it must meet at least one of these criteria:
A) It can be written in decimal form. B) It can be written in binary, octal or hexadecimal form without any repeating digits. C) The sum of its prime factors (excluding the coefficient from 0 itself), when multiplied together equals zero.
Binary numbers do have repeating digits so they cannot represent a rational number; however, we may consider other representations such as octal (base-eight) and hexadecimal (base sixteen). These forms allow us to write all integers that are less than 255 using no more than three digits.
Since any integer less than 255 is finite, it is a rational number and may be written as the sum of its prime factors. For example, in base eight: 34 = 22 + 20; or hexadecimal: 34 = 42 + 32
This means that not only can we write 0 as the sum of its prime factors, we CANNOT include any other digit before our decimal point so this would result in .0 which IS irrational because zero has no repeating digits when represented in octal or hexadecimal form!
Hence 0 is an irrational number – and therefore NOT a rational number!
[ ] Our goal for today’s post was to show proof that there are numbers greater than one that are not rational, which is why there are irrational numbers.
[ ] Tomorrow we will explore the various types of irrational numbers and some methods for finding them!
is 0 a rational number – Yes it is an integer with no repeating digits when represented in octal or hexadecimal form because any number less than 255 can be written as the sum of its prime factors: 34 = 22 + 20; hence 0 IS an irrational number. And therefore NOT a rational one either! We’ll talk more about this tomorrow… No worries though, you don’t need to memorize anything today ;)
The Proof that There Are Numbers Greater Than One That ARE Not Rational (Irrational) Theorem:
is 0 a rational number – The coefficient of the term in the polynomial expression that would produce this representation cannot be expressed as a finite decimal, which means it is not rational. This theorem applies to any representations seen above: octal or hexadecimal notation; units from binary up through septenary (base seven).
The Proof That There Are Numbers Greater Than One That ARE NOT Rational Irrational Number Examples:
is 0 a rational number – An irrational numbers are those numbers where there’s no way to represent them with an infinite series of digits after the decimal point. These include pi and e! Remember these? They’re what we’ll explore more tomorrow ;)
How To Find Irruptive Numbers:
is 0 a rational number – A simple way to find out if the number is irrational or not is by checking for factors. If you can divide it into two equal parts, then it’s an even and therefore, one of its factors; meaning that it IS rational!
is 0 a rationalnumber – Let’s do this with pi (π). π ≈ √(22/71 × 32) = √144421 = 1416+∛464×pi²
This shows us that our original assumption was wrong because there are in fact some numbers greater than one that ARE NOT Rational. It also means we cannot express π as an infinite series of digits after the decimal point.
is 0 a rational number – So, if we keep dividing the decimal point into pi (π), it will eventually end up at some random irrational number since π cannot be written as an infinite series of digits after the decimal. And while this doesn’t mean that all numbers are irrational; they just can’t be expressed in terms of fractions or ratios, but instead have to rely on decimals and exponents!
So for now: is 0 a rational number? Short answer: no ;)