What's the difference between a rational number and an irrational number?

Question

What&#039;s the difference between a rational number and an irrational number?

Maggie Bowman · Accepted Answer

An irrational number is simply a real number that is not rational. In other words, it is a real number that can't be written as the ratio of integers.

A classic example is the number

π

; let's try to get some intuition for irrational numbers through this example. You may be familiar with the first few digits in the decimal expansion of

π

;

\begin{aligned} π = 3.14159 \dots \end{aligned}

Notice that if we truncate this decimal expansion at any point, then the resulting number is rational. Let's say, for instance, that we truncate at the tens place, then we obtain the number 3

3.1

which can be written as

\begin{aligned} 3.1 = \frac{31}{10} \end{aligned}

\begin{aligned} 3.14 & = \frac{314}{100} \\ 3.141 & = \frac{3141}{1000} \\ 3.1415 & = \frac{31415}{10000} \end{aligned}

\begin{aligned} 3.14 & = \frac{314}{100} \\ 3.141 & = \frac{3141}{1000} \\ 3.1415 & = \frac{31415}{10000} \end{aligned}

We can continue in this way, obtaining better and better rational approximations to

π

, but notice that we can never quite get to

π

with only a finite number of decimal digits. We need the full, infinite sequence of decimal digits to get exactly the number

π

. In other words, although there are better and better rational approximations to

π

, each of which is a ratio of integers, there is no way to write exactly the number π as such a ratio.

What's the difference between a rational number and an irrational

Answered question

Answer & Explanation

New Questions in Algebra I