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Compare and order real numbers

Real numbers—including integers, fractions, terminating or repeating decimals, square roots, and familiar irrational numbers—are compared by their positions on the number line, equivalent representations, and appropriate approximations. The reasoning includes ordering negative and positive values, recognizing that a more negative number is smaller, and interpreting symbols such as <, >, and =; it supports inequalities, measurement, and algebra. Complex numbers and more abstract properties of ordered number systems are not included.

Detailed Explanation: Compare and order real numbers

To compare and order real numbers, rewrite them in forms that are easy to compare, such as decimals. Then place them from left to right on a number line.

  • Numbers farther left are smaller.
  • Any negative number is less than any positive number.
  • Among negative numbers, the number with the greater distance from 00 is smaller. For example, −5<−2-5<-2.
  • Use approximations for square roots and other irrational numbers.

Example: Order the numbers from least to greatest:

−2.4,−5,−73,0.3‾,2,1.5-2.4,\quad -\sqrt{5},\quad -\frac{7}{3},\quad 0.\overline{3},\quad \sqrt{2},\quad 1.5

Step 1: Rewrite or approximate the numbers.

−73=−2.3‾≈−2.333-\frac{7}{3}=-2.\overline{3}\approx -2.333 0.3‾=13≈0.3330.\overline{3}=\frac13\approx 0.333 −5≈−2.236-\sqrt5\approx -2.236 2≈1.414\sqrt2\approx 1.414

Now the numbers are approximately:

−2.4,−2.236,−2.333,0.333,1.414,1.5-2.4,\quad -2.236,\quad -2.333,\quad 0.333,\quad 1.414,\quad 1.5

Step 2: Order the negative numbers.

For negative numbers, the more negative number is smaller:

−2.4<−2.333<−2.236-2.4<-2.333<-2.236

So,

−2.4<−73<−5-2.4<-\frac73<-\sqrt5

Step 3: Order the positive numbers.

0.333<1.414<1.50.333<1.414<1.5

So,

0.3‾<2<1.50.\overline3<\sqrt2<1.5

Step 4: Combine the negative and positive numbers.

All negative numbers come before all positive numbers:

−2.4<−73<−5<0.3‾<2<1.5\boxed{-2.4<-\frac73<-\sqrt5<0.\overline3<\sqrt2<1.5}

When using approximations, keep enough decimal places to make the comparison clear.

Learn by doing: Compare and order real numbers

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    Number Types (Rational) - Compare - Negative Fractions, Decimals, Repeating Decimals

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