Lesson 3: Compare and order mixed numbers in various forms.

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

On a number line, for any two numbers $a$ and $b$, if the point representing $a$ is to the left of the point representing $b$, then $a < b$. To plot numbers in various forms, convert them to a common format (like decimals) to determine their position.

Examples

Property

To show the order of a set of numbers, we can write a comparison statement using the symbols for less than ($<$) and greater than ($>$). For a set ordered from least to greatest, the statement is written as $a < b < c$. For a set ordered from greatest to least, the statement is written as $a > b > c$.

Examples