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Compare and order real numbers

Comparing and ordering real numbers means locating rational, irrational, and radical values on a common number line and determining their relative size, including negative numbers, zero, and values written as fractions, decimals, or exact radical forms. Reasoning may involve equivalent forms, benchmarks, place value, estimating radicals, and squaring nonnegative quantities, while avoiding the misconception that a longer decimal representation is necessarily larger; the scope excludes complex numbers and advanced parameterized inequalities.

Detailed Explanation: Compare and order real numbers

To compare real numbers, rewrite them in forms that are easy to place on the same number line. Useful strategies include:

  • Convert fractions to decimals when convenient.
  • Estimate radicals using nearby perfect squares.
  • Remember that numbers farther left on the number line are smaller.
  • Be careful with negatives: multiplying or negating reverses the order.

Example: Order these numbers from least to greatest:

−7,−2.6,−52,0-\sqrt{7},\quad -2.6,\quad -\frac{5}{2},\quad 0

Step 1: Rewrite the fraction.

−52=−2.5-\frac{5}{2}=-2.5

Now compare:

−7,−2.6,−2.5,0-\sqrt{7},\quad -2.6,\quad -2.5,\quad 0

Step 2: Estimate the radical.

Since

2.62=6.76and2.72=7.29,2.6^2=6.76 \quad\text{and}\quad 2.7^2=7.29,

and 77 lies between 6.766.76 and 7.297.29, we know

2.6<7<2.7.2.6<\sqrt{7}<2.7.

Negating reverses the inequalities:

−2.7<−7<−2.6.-2.7<-\sqrt{7}<-2.6.

So −7-\sqrt{7} is less than −2.6-2.6.

Step 3: Place all values from left to right.

Because

−2.6<−2.5<0,-2.6<-2.5<0,

the order from least to greatest is

−7<−2.6<−52<0.\boxed{-\sqrt{7}<-2.6<-\frac{5}{2}<0}.

The approximate number-line locations are −7≈−2.65-\sqrt7\approx-2.65, −2.6-2.6, −2.5-2.5, and 00. Use the values, not the number of digits in a decimal, to decide which number is larger.

Learn by doing: Compare and order real numbers

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    Number Types (Irrational) - Compare - Negative Square Roots, Cube Roots, Pi

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