Table of Contents
Understanding Square Roots
A square root answers this question:
“What number, when multiplied by itself, gives this number?”
If $x^2 = a$, then $x$ is a square root of $a$.
In this course, when we write $\sqrt{a}$, we usually mean the positive square root of $a$ (called the principal square root).
For example:
- $\sqrt{9} = 3$ because $3^2 = 9$.
- $\sqrt{25} = 5$ because $5^2 = 25$.
Note that $(-3)^2 = 9$ as well, so $9$ technically has two square roots: $3$ and $-3$. But by convention, $\sqrt{9}$ means the positive one, $3$.
Square Roots of Perfect Squares
A perfect square is a whole number that is the square of another whole number:
- $1^2 = 1$
- $2^2 = 4$
- $3^2 = 9$
- $4^2 = 16$
- $5^2 = 25$
- $6^2 = 36$
- $7^2 = 49$
- $8^2 = 64$
- $9^2 = 81$
- $10^2 = 100$
So the square roots of these perfect squares are whole numbers:
- $\sqrt{1} = 1$
- $\sqrt{4} = 2$
- $\sqrt{9} = 3$
- $\sqrt{16} = 4$
- $\sqrt{25} = 5$
- $\sqrt{36} = 6$
- $\sqrt{49} = 7$
- $\sqrt{64} = 8$
- $\sqrt{81} = 9$
- $\sqrt{100} = 10$
You do not need to memorize a long list, but it is helpful to be very familiar with the first few squares up to at least $12^2 = 144$.
Notation and the Square Root Symbol
The square root symbol is called the radical sign.
- In $\sqrt{16}$:
- The symbol $\sqrt{\phantom{a}}$ is the radical.
- The number inside, $16$, is called the radicand.
So we can say: “The square root of the radicand $16$ is $4$.”
If you see:
- $\sqrt{a}$: principal (positive) square root.
- $-\sqrt{a}$: the negative of the principal square root.
Example:
- $\sqrt{36} = 6$
- $-\sqrt{36} = -6$
Be careful: the equation
$$x^2 = 36$$
has two solutions:
$$x = 6 \quad \text{or} \quad x = -6.$$
We summarize this as $x = \pm 6$. But
$$\sqrt{36} = 6$$
only gives the positive one.
Square Roots and Area of Squares
Square roots are closely connected to the area of a square.
- If a square has side length $s$, its area is $s^2$.
- If you know the area and want the side length, you take a square root.
If the area is $49$ square units, then the side length is:
$$s = \sqrt{49} = 7.$$
So the square root of the area of a square gives the side length of that square (assuming a positive length).
Square Roots and Negative Numbers
In basic arithmetic with real numbers, no real number squared gives a negative result.
- $3^2 = 9$
- $(-3)^2 = 9$
- In fact, $x^2 \ge 0$ for any real number $x$.
That means expressions like:
- $\sqrt{-1}$
- $\sqrt{-9}$
do not have real-number values in this course. We simply say they are “not real” or “undefined over the real numbers.”
So, for now:
- $\sqrt{a}$ is defined only for $a \ge 0$ in our arithmetic setting.
- For $a < 0$, we say $\sqrt{a}$ is not a real number.
Estimating Square Roots of Non-Perfect Squares
Many numbers are not perfect squares. Their square roots are not whole numbers.
Examples:
- $\sqrt{2}$ is about $1.414\ldots$
- $\sqrt{3}$ is about $1.732\ldots$
- $\sqrt{5}$ is about $2.236\ldots$
To estimate such roots, you can compare with nearby perfect squares.
Example: Estimate $\sqrt{50}$.
- $7^2 = 49$
- $8^2 = 64$
So $50$ is just a little more than $49$, and much less than $64$.
Therefore:
$$7 < \sqrt{50} < 8.$$
Since $50$ is very close to $49$, $\sqrt{50}$ is close to $7$; a rough estimate is $7.1$.
Another example: Estimate $\sqrt{20}$.
- $4^2 = 16$
- $5^2 = 25$
So:
$$4 < \sqrt{20} < 5.$$
Since $20$ is closer to $16$ than to $25$, $\sqrt{20}$ will be a bit more than $4$ but not too close to $5$. A rough estimate is about $4.5$.
Basic Properties Specific to Square Roots
These properties are useful when working with square roots. All numbers here are assumed to be non-negative real numbers.
- $\sqrt{0} = 0$, and $\sqrt{1} = 1$.
- If $a$ and $b$ are non-negative:
$$\sqrt{a} \ge 0,$$
and if $a < b$ then $\sqrt{a} < \sqrt{b}$.
So the square root function preserves order for non-negative numbers. - For a non-negative number $a$:
$$\left(\sqrt{a}\right)^2 = a.$$
This just says that taking a square root and then squaring brings you back. - If $x$ is a real number (possibly negative), then:
$$\sqrt{x^2} = |x|,$$
where $|x|$ is the absolute value of $x$. This happens because $\sqrt{\phantom{a}}$ always gives the non-negative root.
Example:
- $\sqrt{5^2} = \sqrt{25} = 5 = |5|$.
- $\sqrt{(-5)^2} = \sqrt{25} = 5 = |-5|$.
Working With Square Roots in Arithmetic
Within arithmetic, you will often see square roots appear in simple expressions.
Examples of direct evaluation:
- $\sqrt{81} = 9$
- $\sqrt{121} = 11$
- $\sqrt{0} = 0$
Examples involving a negative sign in front:
- $-\sqrt{16} = -4$
- $-2\sqrt{9} = -2 \cdot 3 = -6$
Examples combining squares and square roots:
- $\left(\sqrt{49}\right)^2 = 49$
- $\sqrt{7^2} = |7| = 7$
- $\sqrt{(-7)^2} = | -7 | = 7$
Common Mistakes to Avoid
- Dropping the “positive only” idea of $\sqrt{\phantom{a}}$
Remember $\sqrt{a}$ is defined as the positive square root when $a > 0$. - Correct: $\sqrt{36} = 6$
- Incorrect: $\sqrt{36} = \pm 6$ (this notation belongs to solving equations like $x^2 = 36$).
- Taking square roots of negative numbers in basic arithmetic
- In this course level: $\sqrt{-4}$ is “not a real number,” not $-2$.
- $(-2)^2 = 4$, not $-4$.
- Confusing $\sqrt{a^2}$ with $a$ instead of $|a|$
- $\sqrt{5^2} = 5$, but
- $\sqrt{(-5)^2} = 5$, not $-5$.
- Assuming non-perfect squares have “nice” square roots
- $\sqrt{2}$ is not a simple fraction like $1.4$ or $3/2$; its decimal goes on without repeating.
- For many numbers, the square root is an approximation unless it is a perfect square (or can be simplified in more advanced ways).
Practice Ideas
To become comfortable with square roots, you might:
- Memorize squares of the integers from $1$ to $15$ and use them to quickly find square roots of perfect squares.
- Given a number like $n$, decide whether it is a perfect square by seeing if it matches any of $1^2, 2^2, 3^2, \dots$.
- For numbers that are not perfect squares, use nearby perfect squares to estimate the square root to the nearest whole number.