Problem 11 - Algebra 1: Concepts and Skills

Question

Determine whether each expression is rational or irrational. $$ \sqrt{10} $$

Step-by-Step Solution

Answer

\(\sqrt{10}\) is an irrational number because its decimal representation neither terminates nor repeats.

Step 1: Understanding \(\sqrt{10}\)

We first need to understand what the square root is. The square root of a number 'n' is any number that, when multiplied by itself, gives 'n'. So, when we look at \(\sqrt{10}\), we're asking 'what number, when multiplied by itself, equals 10?'.

Step 2: Calculate the Square Root

Calculate \(\sqrt{10}\) using a calculator. The result is approximately 3.16227766017 which extend indefinitely without any repeating pattern.

Step 3: Determine if it's Rational or Irrational

Now, having calculated \(\sqrt{10}\), we notice that its decimal representation neither terminates nor repeats, which means it cannot be expressed precisely as a fraction. Thus, \(\sqrt{10}\) is irrational.

Key Concepts

Square Root

The square root is a fundamental concept in mathematics. When you see the symbol \(\sqrt{n}\), it signifies a number that, when multiplied by itself, results in \(n\). For example, if you take \(\sqrt{10}\), it is asking for a number which, when squared (multiplied by itself), gives us 10. However, determining the exact square root of numbers that are not perfect squares (like 10) is not straightforward.

• Perfect Squares: Numbers like 1, 4, 9, 16, where their square roots are whole numbers.
• Non-Perfect Squares: Numbers like 2, 3, 10, where calculating an exact square root gives a non-repeating, non-ending decimal.

In the case of \(\sqrt{10}\), when calculated, it provides an endless sequence of decimals without a predictable pattern. This non-repeating, non-terminating nature is key to identifying whether a square root results in a rational or an irrational number.

Decimal Representation

Decimal representation is how numbers are expressed in a base-10 system using digits from 0 to 9. For many numbers, this system allows us to express them neatly and clearly, such as with integers or terminating decimals like 1.5 or 2.75.
However, with numbers such as square roots, the decimal representation can be more complex. Numbers like \(\sqrt{10}\) will have decimals that continue indefinitely and do not repeat a pattern.

• Terminating Decimals: These numbers stop after a certain number of digits, like 0.75 or 1.5.
• Repeating Decimals: These numbers have a digit or block of digits that repeat infinitely, like 0.333... or 0.142857...
• Non-Terminating, Non-Repeating Decimals: These decimals, like the one for \(\sqrt{10}\) (approximately 3.16227766017...), go on forever without forming a repeating pattern.

Understanding these forms of decimal representation helps us classify numbers into rational and irrational, based on their ability to terminate or repeat.

Fractions

Fractions are a way to express a number as part of a whole. They are written in the form of \(\frac{a}{b}\) where "a" is the numerator and "b" is the denominator. A key property of rational numbers is their ability to be expressed as a fraction.
Rational numbers are those which can be converted precisely into a fraction. For example, \(0.5\) can be represented as \(\frac{1}{2}\), and \(0.333...\) can be written as \(\frac{1}{3}\).

• Types of Numbers Expressed as Fractions: Integers, terminating decimals, and repeating decimals.
• Irreducible Fractions: Fractions in their simplest form where the greatest common divisor of the numerator and the denominator is 1, e.g., \(\frac{3}{4}\).

However, irrational numbers, such as \(\sqrt{10}\), cannot be neatly expressed as fractions due to their complex and non-repeating decimal nature. This distinction makes it clear why something like \(\sqrt{10}\) is classified as irrational.