Some of the Coolest Numbers Are Irrational

At Mathnasium of Littleton some of our favorite numbers are irrational. We celebrate “pi day” on March 14th each year with pie and activities related to this irrational number, but we like other irrational numbers too!

What is an Irrational Number?
An irrational number is a number that can’t be expressed as a ratio between two numbers. It is number where the digits to the right of the decimal go on indefinitely without a repeating pattern. That means whole numbers are never irrational numbers because the only number after the decimal would be 0. We like the idea that irrational numbers go on forever. That just boggles the mind.

4 Famous Irrational Numbers
1)   Pi describes the ratio of a circle’s circumference in relationship to its diameter. For basic calculations the number 3.142 is used for pi.
2)   Phi Φ (pronounced fie) describes the ratio of line segments divided in a specific way. It is known as the “the Golden Number” because the geometric relationships are seen in art, nature, architecture, physics, theology, and more! Some people may relate “the Golden Number” to the Fibonacci sequence. In fact there are mathematical correlations between Fibonacci sequence and Φ. For basic calculations the number 1.618 is used for Phi. We like Phi because it has two properties that no other positive number has. Its square is one greater than itself. Φ² = Φ + 1 and  the reciprocal (1 divided by the number) of Phi is one less than itself. 1/Φ=Φ-1
3)   The square root of 2 is another well-known irrational number. The approximate square root of 2 is 1.414. We like this number because you can prove it is irrational.
4)   The number e or Euler’s number, is the base of natural logarithms. The approximate value of e is 2.718. We like e because it is used for calculating continuous compounding interest such as for calculating the return on an investment. Who doesn’t love making money? For those not satisfied with just going out three places past the decimal point, here is a link to the first five million digits.

Are there Practical Uses for Irrational Numbers?
Scientists, accountants, and engineers rely on e for much of their work. They either leave the number in the symbolic form or decide how accurate (how many digits) they need to use.

Irrational but Fun
Just thinking that irrational numbers exist sparks some interesting questions. They provide math lovers, like us, some fascinating creative exercises, like proving that a number is irrational. They provide people who are hesitant about math some interesting things to think about.

Here are a few questions about irrational numbers to spark your interest.

How did early mathematicians discover that irrational numbers exist?

At what point did they give up and decide that the numbers to the right of the decimals would never repeat and never stop?

How far did they calculate out before making that decision … 1000 places ….10, 000 places?

How many irrational numbers are there? One might think that there would be more rational numbers than irrational numbers, but the opposite is true. Numbers are infinite so there are infinite amounts of both rational and irrational numbers. Rational numbers are theoretically countable and irrational numbers are not, so some mathematicians say there are more irrational numbers than rational numbers. That means that some sets of infinity are bigger than other sets of infinity! How is that for a mind boggling thought!

Can you put irrational numbers in fraction form? No! You cannot express rational numbers as a fraction because fractions are ratios between two numbers.  33.3333333333 … with the 3s repeating is still a rational number. It can be written as 100/3. That’s why irrational numbers are usually expressed as a symbol like π, Φ, and e. It is the only accurate way to write them.

Mathematically speaking, are irrational numbers “real” or “imaginary”? They are real. There is a whole other set of numbers known as “imaginary” and yet they still exist. Imaginary numbers can’t actually be described as rational or irrational. That’s a topic for another day.

Irrational numbers and Mathnasium of Littleton
We love thinking about advanced math. You don’t have to be in an advanced math course to start thinking about advanced math topics. If you want more articles to spark your math creativity check out:

A Mathematical Look at Snowflakes
6 Ways for Accelerated and Advanced Math Students to Get Excited About Math Again
10 Fun Ways to Incorporate Math During School Breaks
The Long Car Ride
Why Does Mathnasium of Littleton Love Math Games?
I’m Never Going to Use Calculus After High School
Math Books Beyond 1,2,3
Family Friendly Math Movies
Math Skills to Learn from a 7-Year-Old
Mathematician Spotlight Katherine G. Johnson
The Number Zero
The Great Tradition of Arithmetic

Got a math lover in your house? Bring them on into Mathnasium of Littleton. They will be given the freedom to pursue math just for fun and at their own pace. Also read Top 5 Math Classes to Take for 21st Century Success and Careers in Math

Got a math hater in your house? That’s an emergency!!! Please read Is Hating Math Normal? Then bring them in so we can help get them on track.

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