Lesson 4: Approximating Square Roots

Property

The set of natural numbers includes the numbers used for counting: {1, 2, 3, …}.
The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3, …}.
The set of integers adds the negative natural numbers to the set of whole numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}.
The set of rational numbers includes fractions written as $\{\frac{m}{n} | m \text{ and } n \text{ are integers and } n \neq 0\}$.
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: {$h$ | $h$ is not a rational number}.

Examples

• Classify $\sqrt{64}$: This simplifies to $8$. So, it is a natural number (N), whole number (W), integer (I), and rational number (Q).
• Classify $\frac{14}{3}$: As a fraction of integers, it is a rational number (Q). As a decimal, it is $4.666...$, which is a repeating decimal.
• Classify $\sqrt{13}$: This cannot be simplified to a whole number or a fraction of integers, so it is an irrational number (Q').

Explanation

Think of number sets like nesting dolls. Naturals fit inside wholes, which fit inside integers, which fit inside rationals. Irrationals are a separate group, and all of them together form the real numbers.

Property

When a positive integer is not a perfect square, its square root is an irrational number. An irrational number cannot be written as the ratio of two integers, and its decimal form does not terminate or repeat.

Examples

Property

To approximate an irrational square root $\sqrt{n}$ to the nearest integer, find the two consecutive perfect squares that $n$ is between. If $a^2 < n < (a+1)^2$, then the value of $\sqrt{n}$ is strictly between the integers $a$ and $a+1$:

Examples

• Approximate $\sqrt{30}$ to the nearest integer.

The perfect squares closest to $30$ are $25$ and $36$.

Since $30$ is closer to $25$ than to $36$, the best integer approximation is $5$. So, $\sqrt{30} \approx 5$.

• Approximate $\sqrt{85}$ to the nearest integer.

The perfect squares closest to $85$ are $81$ and $100$.

Since $85$ is closer to $81$ than to $100$, the best integer approximation is $9$. So, $\sqrt{85} \approx 9$.