1. Remember the sets of numbers

• Natural numbers : N = { 0 , 1 , 2 , 3 , 4 , ... }
• Integers : Z = { ... , -2 , -1 , 0 , 1 , 2 , ... }
• Rational numbers : Q = { a/b : a $\in$ Z , b $\in$ Z , b $\neq$ 0 }

Examples : 3/4 , -1/2 , 0 , 5 , 0.2 , 0.7 (repeating) , 13% , $\sqrt{36}$ , ∛-64 , ...

• Irrational numbers : Q' (Q with a dash) — numbers that CANNOT be written in the form a/b where a , b are integers and b $\neq$ 0.
• Real numbers : R = Q $\cup$ Q' (rationals union irrationals).

2. How to tell rational from irrational

3. Special irrational numbers

• $\pi$ (pi) : the ratio between the circumference of a circle and its diameter. $\pi$ ≈ 3.141592654...
• $\phi$ (golden ratio) : a ratio that appears in nature. $\phi$ ≈ 1.618033989...
• Other examples : $\sqrt{5}$ + 1 , 1 - $\sqrt{7}$ , 2$\sqrt{7}$ , $\sqrt[3]{9}$ , $\sqrt{5}$ + $\sqrt[3]{9}$ , 1/$\sqrt{7}$ , -$\sqrt{11}$ , $\sqrt{6}$ , ...

4. Location on the number line

• Any irrational number lies between two rational numbers, and can be represented on the number line by a point that lies between them.
• To find two consecutive integers between which an irrational number lies, compare it with known squares/cubes.

Example : $\sqrt{13}$ lies between 3 and 4 because 9 < 13 < 16.

5. Venn diagram relationships

• N $\subset$ Z $\subset$ Q $\subset$ R
• Q' $\subset$ R
• Q $\cap$ Q' = $\emptyset$ (empty set) — a number is either rational OR irrational, not both.

6. REMEMBER

• There is NO square root of a negative real number : √-4 $\notin$ R (not a real number).
• The cube root of a negative number IS a real number : ∛-8 = -2 $\in$ R.

7. Solving equations with roots

• Equation of the form x³ = a : x = $\sqrt[3]{a}$
• Equation of the form x² = a (a > 0) : x = +$\sqrt{a}$ or x = -$\sqrt{a}$
• Example : x³ - 11 = 5 $\rightarrow$ x³ = 16 $\rightarrow$ x = $\sqrt[3]{16}$. Example : 2x³ + 13 = 67 $\rightarrow$ x³ = 27 $\rightarrow$ x = 3.
• A product is zero when at least one factor is zero : (x² - 9)(x³ - 5) = 0 $\rightarrow$ x² = 9 or x³ = 5.

8. WORKED EXAMPLES

Example 1 : Which is rational and which is irrational? State why.

• $\sqrt{25}$ = 5 $\rightarrow$ rational (perfect square).
• $\sqrt{13}$ $\rightarrow$ irrational (13 is not a perfect square).
• 3.6 $\rightarrow$ rational (terminating decimal).
• 0.323232... $\rightarrow$ rational (repeating decimal).
• $\sqrt[3]{16}$ $\rightarrow$ irrational (16 is not a perfect cube).
• $\sqrt{9}$ + $\sqrt{16}$ = 3 + 4 = 7 $\rightarrow$ rational.

Example 2 : Put $\in$ or $\notin$ : $\sqrt{5}$ ___ R

• $\sqrt{5}$ is a real number $\rightarrow$ $\sqrt{5}$ $\in$ R.

Example 3 : Find two consecutive integers for $\sqrt{5}$.

• 4 < 5 < 9 $\rightarrow$ $\sqrt{4}$ < $\sqrt{5}$ < $\sqrt{9}$ $\rightarrow$ 2 < $\sqrt{5}$ < 3. So $\sqrt{5}$ lies between 2 and 3.