Real numbers that cannot be represented as a ratio are known as irrational numbers. Whilst rational numbers are also real numbers, they are different from irrational numbers. In the 5th century BC, Hippasus, a Pythagorean philosopher, discovered irrational numbers. Keep reading to learn more about irrational numbers and the differences between irrational and rational numbers.
What are irrational numbers?
An irrational number’s decimal expansion is neither terminating nor repeating. For example, 2.59265… does not terminate so it’s an irrational number. Irrational numbers are real numbers that cannot be expressed as a simple fraction. They can’t be stated as a ratio like p/q, where p and q are both integers, and q ≠ 0.
What are terminating and repeating decimals?
Even though a repeating decimal has an infinite number of digits, all of them are known. The digits after the decimal point cannot all be 0 for the decimal to be considered recurring. For non-terminating decimals that do not repeat, not all of the digits are known. No matter how many numbers are known, there will always be a digit after it that needs to be determined.
It’s worth mentioning that 1/3 is a repeating decimal as well as a non-terminating decimal. It’s crucial to grasp the differences between rational and irrational decimals to distinguish them. All decimals that terminate in a digit are rational numbers. Both ending and recurring decimals can be expressed using a fraction form.
Non-terminating and non-repeating irrational numbers, the most notable of which is pi, exist. Two instances are pi (3.14159…) and the square root of 2 (1.4142135…). No matter how many digits we compute, neither will ever finish or repeat.
What are rational numbers?
In arithmetic, rational numbers are a sort of number that we normally learn after integers. Rational numbers can be represented as a quotient of two whole numbers. They are expressed as a fraction a / b, where a and b are integers and b is different from zero.
Whilst whole numbers make up fractions, such as 2 being the same as 2/1, an integer is the notation within the fraction, i.e., the 2 within 2/1. It can be slightly confusing but know that integers is an umbrella term that encompasses all numbers.
Properties of irrational numbers
Irrational number properties assist us in identifying irrational numbers among a group of real numbers:
- Non-terminating and non-recurring decimals make up irrational numbers.
- Only real numbers are used.
- The outcome of an irrational number x and a rational number y is an irrational number.
- The product of any irrational number multiplied by any nonzero rational number is an irrational number. The product of an irrational number x with a rational number y is irrational.
- The least common multiple (LCM) of any two irrational numbers may or may not exist.
- Two irrational numbers added, subtracted, multiplied, and divided may or may not be rational numbers.
How do you know if a number is irrational?
Rational numbers can be stated in the form of a ratio or fraction. Fractions cannot be used to represent irrational numbers. If a number can be written or translated into the p/q form, where p and q are integers and q is a non-zero number, it is said to be rational. If it cannot, it is said to be irrational.
Explanation
Any number that can be represented or written in the p/q form, where p and q are integers and q is a non-zero number, is a rational number.
Example: 12/5, -9/13, 8/1
On the other hand, an irrational number cannot be stated in p/q form, and its decimal expansion is non-repeating and non-terminating.
Example: √2, √7, √11
We can recognize and categorize numbers as rational or irrational using these definitions. The p/q form is key to defining and categorizing rational and irrational numbers. If the number fits the p/q form then it is rational, if it cannot then it is irrational.
Irrational number symbol
Let’s look at the symbols for various types of numbers before learning about irrational numbers.
- N stands for natural numbers
- I stands for imaginary numbers
- R stands for real numbers
- Q stands for rational numbers
Both rational and irrational numbers make up real numbers. Irrational numbers can be obtained by subtracting rational numbers (Q) from real numbers, as defined by (R-Q) (R). It’s also possible to write it as (R\Q).
Rational vs. irrational numbers
A rational number is any number that can be expressed as a ratio or as a fraction (p/q). The numerator (p) and denominator (q), when q is not zero, may be included. A whole number or an integer can be a rational number.
Take 2/3 = 0.6666 = 0.67 for example. Because the decimal value is repeated we calculated 0.67. √4 equals 2 and -2, where 2 is a positive integer, and -2 is a negative integer.
Rational
It can be written as a fraction or ratio, such as p/q, where q is less than zero. The decimal expansion is recurring and either ending or non-terminating (repeating).
Example: 0.33333, 0.656565.., 1.75
Irrational
It cannot be stated as a fraction or as a ratio. At any point in time, the decimal expansion is non-terminating and non-recurring.
Example: π, √13, e
Irrational number examples
Which of the following is an irrational number?
2, √16, 1/2, √5
Irrational numbers are real numbers that cannot be expressed as a simple fraction nor will their decimal expansion terminate or repeat. As such, let’s look through the numbers stated:
- 2 is a whole number so it is not an irrational number.
- √16 looks like it could be an irrational number, but the square root of 16 is 4 without any remainders. Therefore, it is not irrational.
- 1/2 expressed as a decimal is 0.5 which terminates and does not repeat. Therefore, it is also not an irrational number.
- √5 is an irrational number as the square root is 2.2360679774997896964091736 which does not terminate.
Which of the following is the largest irrational number?
π, √5, 4.64378123…, √21
Converting all these into decimals will allow us to compare and contrast these numbers. As these are all irrational numbers resulting in an infinite number of decimal numbers, we will be confining them to 3 decimal places.
- π as a decimal is 3.143.
- √5 is 2.236
- 4.644 can stay as it is.
- √21 is 4.582
From looking at all the decimals above, we can compare them to determine which one is the largest.
From largest to smallest, the irrational numbers rank:
4.64378123…, √21, √5, π
Is 7/12 irrational?
To figure this out we have to convert this fraction into a decimal:
0.5833…
12 ) 7.0000
60
100
96
40
36
40
36
4
As such, 7/12 = 0.58333…
This is a non-terminating decimal meaning that 7/12 is an irrational number.
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