Rational vs. Irrational Numbers: What’s the Difference? 5 Facts

Math is really just a game with numbers. A number is an arithmetic value that can be a figure, word, or symbol that stands for a quantity. It can be used for counting, measuring, calculating, labelling, and many other things.

A number can be a natural number, a whole number, an integer, a real number, or a complex number. There are two more types of real numbers: rational and irrational. Rational numbers are like whole numbers and fractions.

On the other hand, irrational numbers are numbers that can’t be written as a fraction. In this post, we will look at how rational and irrational numbers are different. Check it out.

Definition of Rational Numbers

The word ratio is where the word ratio comes from. A ratio is a simple fractional comparison of two amounts.

A number is rational if it can be written as a fraction like p/q, where both p and q are integers and q is a natural number (a non-zero number).

Rational numbers include integers, fractions (even mixed fractions), decimals that repeat, and decimals that have a limit, among other things.

Rational vs. Irrational Numbers

Examples of Rational Number

  • 1/9 – Both numerator and denominator are integers.
  • 7 – Can be expressed as 7/1, wherein 7 is the quotient of integers 7 and 1.
  • √16 – As the square root can be simplified to 4, which is the quotient of fraction 4/1
  • 0.5 – Can be written as 5/10 or 1/2 and all terminating decimals are rational.
  • 0.3333333333 – All recurring decimals are rational.

Definition of Irrational Numbers

Irrational means that a number can’t be broken down into any fraction of an integer (x) or a natural number (y). It can also be seen as a number that doesn’t make sense. The irrational number’s decimal expansion is neither limited nor repeated.

It is made up of weird and strange numbers like pi (the most common irrational number) and e. A surd is a square or cube that isn’t perfect and can’t be made smaller to get the square or cube root.

Examples of Irrational Number

  • √2 – √2 cannot be simplified and so, it is irrational.
  • √7/5 – The given number is a fraction, but it is not the only criteria to be called as the rational number. Both numerator and denominator need to integers and √7 is not an integer. Hence, the given number is irrational.
  • 3/0 – Fraction with denominator zero, is irrational.
  • π – As the decimal value of π is never-ending, never-repeating and never shows any pattern. Therefore, the value of pi is not exactly equal to any fraction. The number 22/7 is just and approximation.
  • 0.3131131113 – The decimals are neither terminating nor recurring. So it cannot be expressed as a quotient of a fraction.

Key Differences Between Rational and Irrational Numbers

There are clear ways to tell the difference between rational and irrational numbers:

  • A rational number can be written as the ratio of two whole numbers. Irrational numbers can’t be written as the ratio of two whole numbers.
  • Rational numbers have a whole number in both the numerator and the denominator, and the denominator is not equal to zero. On the other hand, an irrational number can’t be written as a fraction.
  • Rational numbers include numbers like 9, 16, 25, and so on that are perfect squares. Irrational numbers are things like 2, 3, 5, and so on that don’t make sense.
  • The rational number only has decimals that are finite and repeat. On the other hand, irrational numbers are those whose decimal expansion goes on forever, doesn’t repeat, and has no pattern.

Comparison Chart

BasisRational NumberIrrational Number
MeaningA rational number can be written as a ratio involving two integers.Irrational numbers cannot be expressed as any ratio involving two integers.
FractionsRational numbers can be expressed as a fraction. The denominator of a fraction cannot be equal to zero.Irrational numbers can’t be expressed as a fraction and will always have a denominator that is equal to zero
IncludesA rational number contains perfect squaresSurds are included in irrational numbers.
Decimal ExpansionRational numbers have recurring or finite decimals.Rational numbers can have a terminating or repeating decimal, whereas irrational numbers cannot.Rational numbers are based on fractions, which always have a finite representation.All rational numbers are real numbers. However, not all real numbers are rational.Irrational numbers have non-recurring or non-finite decimals.Irrational numbers are not based on fractions and have limitless decimal expansion.There are infinitely more irrational numbers than there are rational numbers.

Why Do We Use the Words ‘Rational’ and ‘Irrational’?

Why do we say that things are rational or irrational? That doesn’t look very clear. Koloczyk says that “rational” is often used to mean “based on reason” or something similar.

It seems to have been used for math in British writings as early as the 1200s (per the Oxford English Dictionary). If you look at the Latin roots of the words “rational” and “ratio,” you’ll see that they both mean “reasoning” in a broad sense.

Both rational and irrational numbers have been very important to the development of civilization. Mark Zegarelli, a math tutor and author who has written 10 books in the “For Dummies” series, says that numbers came much later than language.

He thinks that hunters and gatherers didn’t need to be very precise with numbers, as long as they could roughly estimate and compare amounts.

Zegarelli says, “They needed to know things like, ‘We don’t have any more apples.'” “We have exactly 152 apples” was information they didn’t need.

But when people started carving out pieces of land to make farms, build cities, and make and trade goods, they needed more complicated math.

Kolaczyk says, “Let’s say you build a house with a roof that is the same height as the ground at its highest point.” How far does the surface of the roof go from the top to the edge? The rise is always a square-root-of-two factor (run). That number is also not logical.

Carrie Manore says that even though technology has changed a lot in the 21st century, irrational numbers are still important.

She is a scientist and mathematician at the Los Alamos National Laboratory. She works in the Information Systems and Modeling Group.

“Pi is a natural place to start talking about irrational numbers,” Manore writes in an email. “It is necessary to know how to figure out a circle’s area and circumference.”

It is needed to figure out angles, which are important for navigation, architecture, surveying, engineering, and many other things.

Sines and cosines, both of which involve pi, are used to send radio waves.” Also, irrational numbers are a big part of the complicated math that makes high-frequency stock trading, modelling, forecasting, and most statistical analysis possible. All of these things are important to the way our society works.

You could keep going on and on. “In fact, in today’s world, it’s almost more natural to ask where irrational numbers aren’t used.” Manore tells us.

Frequently Asked Questions

Are all integers rational numbers?

Yes, a rational number is any number that can be written as a fraction. All numbers fit into this definition.

Are negative numbers rational?

Yes, most negative numbers make a lot of sense. A rational number is any number that can be written as a fraction.

Some examples of numbers are whole numbers, fractions, decimals that end, and decimals that repeat. Logic isn’t much affected by whether you’re happy or sad.

Are all rational numbers whole numbers?

No, not all whole numbers are rational numbers. Rational numbers are all the numbers that end or repeat.

Rational vs. Irrational Numbers

A whole number is any number that is greater than zero or equal to zero and has no fractional parts. For example, 2.7 is not a whole number, but it is a rational number.

What is the difference between rational and irrational numbers?

A rational number can be written as an exact fraction, but an irrational number can’t. A rational number is any whole number, fraction, or decimal that ends or keeps coming back.

Any number that can’t be turned into a fraction and doesn’t fit the definition of a rational number is an irrational number.

Conclusion

Unlike rational numbers, which can be written as fractions, irrational numbers can only be written in decimal form. Also, while every whole number is a rational number, not every number that is not a whole number is irrational.

 

The difference between rational and irrational numbers is important for anyone who works with fractions and decimals.

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