Operations on Real Numbers

Numbers exist to be combined: added, subtracted, multiplied, divided. With rationals this is routine. With irrationals it gets interesting , the sum of two irrationals might be rational, or might not. This lesson collects all the closure rules and surd manipulations you need for the rest of high school maths.

Definitions

A surd is an expression involving a root that does not simplify to a rational number , typically $\sqrt{a}$ where $a$ is a positive rational whose root is irrational. Examples: $\sqrt{2}, \sqrt[3]{5}, 2\sqrt{7}$.

Two surds are like if they have the same radical, e.g. $3\sqrt{2}$ and $-7\sqrt{2}$. They are unlike if the radicals differ, e.g. $\sqrt{2}$ and $\sqrt{3}$. Like surds combine like algebraic terms; unlike surds do not.

Concept and rules

Closure of rationals. If $a, b$ are rational, then $a + b, a - b, a \cdot b$, and (if $b \ne 0$) $\dfrac{a}{b}$ are all rational. The rationals are closed under all four basic operations.

Mixing rational and irrational.

• Rational $\pm$ irrational $=$ irrational. (If $r + i$ were rational, then $i = (r+i) - r$ would be rational, contradicting irrationality.)
• (Non-zero rational) $\times$ irrational $=$ irrational. Same reason.
• (Non-zero rational) $\div$ irrational $=$ irrational. Same.

Irrational with irrational , no clean rule. Anything can happen.

• $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$, irrational.
• $\sqrt{2} + (1 - \sqrt{2}) = 1$, rational.
• $\sqrt{2} \cdot \sqrt{2} = 2$, rational.
• $\sqrt{2} \cdot \sqrt{3} = \sqrt{6}$, irrational.

So when combining two irrationals you must compute carefully , the result is irrational only by accident, not by rule.

Adding and subtracting surds. Combine like surds the way you combine like terms: $3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}, \qquad 8\sqrt{3} - 2\sqrt{3} = 6\sqrt{3}.$ Unlike surds do not combine: $\sqrt{2} + \sqrt{3}$ stays $\sqrt{2} + \sqrt{3}$. A common trick is to simplify first: $\sqrt{12} - \sqrt{3} = 2\sqrt{3} - \sqrt{3} = \sqrt{3}$.

Multiplying surds. For $a, b \ge 0$, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, \qquad \dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}} \ (b \ne 0).$ These are special cases of the laws of exponents we will meet next. So $\sqrt{6} \cdot \sqrt{8} = \sqrt{48} = 4\sqrt{3}$.

Distributive rule. Treat a surd like a variable: $(\sqrt{2} + 3)(\sqrt{2} - 3) = (\sqrt{2})^2 - 3^2 = 2 - 9 = -7$. That last identity, $(a+b)(a-b) = a^2 - b^2$, is the engine behind rationalisation (next lesson).

Powers of surds. $(\sqrt{a})^2 = a$ for $a \ge 0$. So $(\sqrt{2})^4 = ((\sqrt{2})^2)^2 = 4$, and $(\sqrt{3})^5 = (\sqrt{3})^4 \cdot \sqrt{3} = 9\sqrt{3}$.

Simplifying surds. Pull perfect-square factors out of the radical: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}, \qquad \sqrt{50} - \sqrt{18} = 5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}.$ Always simplify before adding or comparing surds , many "unlike" surds become "like" after simplification.

Worked examples

Example 1. Simplify $\sqrt{45} + \sqrt{20} - \sqrt{5}$.

$\sqrt{45} = 3\sqrt{5}$, $\sqrt{20} = 2\sqrt{5}$. So expression $= 3\sqrt{5} + 2\sqrt{5} - \sqrt{5} = 4\sqrt{5}$.

Example 2. Compute $(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})$.

By $(a+b)(a-b) = a^2 - b^2$: $(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$.

Example 3. Simplify $(2 + \sqrt{5})(2 - \sqrt{5})$ and decide if the result is rational.

$= 2^2 - (\sqrt{5})^2 = 4 - 5 = -1$. Rational.

Example 4. Simplify $\sqrt{8} \cdot \sqrt{12}$.

$\sqrt{8 \cdot 12} = \sqrt{96} = \sqrt{16 \cdot 6} = 4\sqrt{6}$.

Example 5. Is $(\sqrt{2} + \sqrt{3})^2$ rational or irrational?

$(\sqrt{2})^2 + 2\sqrt{2}\sqrt{3} + (\sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6}$. Since $\sqrt{6}$ is irrational, $2\sqrt{6}$ is irrational, and rational $+$ irrational is irrational. So the expression is irrational.

Try it yourself

1. Simplify $\sqrt{50} + \sqrt{32} - \sqrt{8}$.
2. Multiply: $(3 + \sqrt{2})(3 - \sqrt{2})$.
3. Simplify $\sqrt{18} \cdot \sqrt{50}$.
4. Compute $(2\sqrt{3} - 3\sqrt{2})^2$.
5. Decide whether each is rational or irrational: $\sqrt{7} \cdot \sqrt{28}$, $\sqrt{2} + 0$, $(\sqrt{5} + 1)(\sqrt{5} - 1)$, $\sqrt{2} \cdot \sqrt{3}$.
6. Simplify $\sqrt{45} - \sqrt{20} + \sqrt{5}$.
7. If $x = \sqrt{2} + 1$ and $y = \sqrt{2} - 1$, find $x + y$, $x - y$, $xy$ and $\dfrac{x}{y}$ (the last needs rationalisation , leave it as a fraction).
8. Find a rational number whose product with $\sqrt{3}$ is irrational.
9. Find two irrationals whose sum is the rational number $7$.
10. Is $\sqrt{2} \cdot \sqrt{8}$ rational? Justify.

Pitfalls / Insight

• $\sqrt{a} + \sqrt{b} \ne \sqrt{a+b}$. Try $a = b = 1$: $1 + 1 = 2$, but $\sqrt{2} \ne 2$.
• Always simplify before checking "like" or "unlike". $\sqrt{8}$ and $\sqrt{2}$ look unlike but $\sqrt{8} = 2\sqrt{2}$.
• A surd is just a number. Treat $\sqrt{2}$ like a variable for algebra; only at the very end ask whether the answer is rational.

Insight. When you see $\sqrt{a} + \sqrt{b}$ or $\sqrt{a} - \sqrt{b}$, immediately think of their conjugate partner. Their product is $a - b$, with no surd left. That single observation kills most "rationalisation" questions instantly.