Lesson 13-4: Compare and Order Rational and Irrational Numbers

Property

To compare and order real numbers presented in different forms (fractions, mixed numbers, square roots, or $\pi$), convert them all into a common decimal format. For the irrational number $\pi$, use the approximations 3.14 or $\frac{22}{7}$. Once converted, compare the decimals digit by digit from left to right, starting with the greatest place value.

Examples

• Compare $\frac{3}{4}$, 0.8, and $\sqrt{2}$:

Convert to decimals: $0.75$, $0.8$, and $1.414...$
Comparing the tenths place, $\frac{3}{4} < 0.8 < \sqrt{2}$.

• Order $-\frac{5}{3}$, -1.8, and $-\sqrt{3}$ from least to greatest:

Convert to decimals: $-1.667...$, $-1.8$, and $-1.732...$
Remember that for negative numbers, the number with the greater absolute value is smaller: $-1.8 < -\sqrt{3} < -\frac{5}{3}$.

• Compare $\frac{22}{7}$ and $\pi$:

$\frac{22}{7} = 3.142857...$ and $\pi = 3.141592...$
Comparing the thousandths place, $\pi < \frac{22}{7}$.

Explanation

It is incredibly difficult to directly compare numbers when they are wearing different "outfits" (like a mixed number vs. a square root). Decimals act as a universal translator! By converting all numbers to decimals, you create a level playing field where you can easily compare them digit by digit.

Property

To order a mixed set of rational and irrational numbers visually, approximate the irrational numbers as decimals and convert any fractions. Then, plot all numbers on a number line to determine their relative positions. Numbers always increase in value from left to right on the number line.

Examples

• Order from least to greatest: 2.5, $\sqrt{7}$, $\frac{8}{3}$

Since $\sqrt{7} \approx 2.646$ and $\frac{8}{3} \approx 2.667$, the relative order is $2.5 < \sqrt{7} < \frac{8}{3}$.

• Order from least to greatest: $\sqrt{15}$, 4, 3.9, $\sqrt{14}$

Since $\sqrt{15} \approx 3.873$ and $\sqrt{14} \approx 3.742$, the sequence is $\sqrt{14} < \sqrt{15} < 3.9 < 4$.

Explanation

Property

When comparing real numbers, avoid premature rounding and false linear scaling assumptions:

1. Precision: Expand decimal approximations to sufficient places (at least one more digit than the number you are comparing against) to avoid false equalities.
2. Non-Linear Scaling: Square roots do not scale linearly. For example, doubling the input does not double the output.

Examples

• Error 1 (Premature Rounding): Compare $\sqrt{10}$ and 3.16.

If you round $\sqrt{10}$ to one decimal place (3.2) too early, you might think it equals 3.16 (which also rounds to 3.2). Using full precision ($\sqrt{10} \approx 3.162...$) reveals the truth: $\sqrt{10} > 3.16$.

• Error 2 (Scaling Illusion): Compare $\sqrt{50}$ and $2\sqrt{25}$.

A common mistake is assuming they are equal because $50 = 2 \cdot 25$. However, evaluating them shows $\sqrt{50} \approx 7.07$ and $2\sqrt{25} = 2 \cdot 5 = 10$. Therefore, $\sqrt{50} < 2\sqrt{25}$.

Explanation

When ordering numbers that are very close in value, rounding too early is a trap that hides small, crucial differences. Always keep an extra decimal place! Additionally, operations like square roots do not behave like simple multiplication. Calculating the exact decimal approximation for each individual term is the only foolproof way to prevent these comparison mistakes.