Comparing Negative Rational Numbers

Greater means farther right

On a number line, values increase as you move to the right. Every negative number lies to the left of zero. Among negative numbers, the one closer to zero lies farther right and is therefore greater. Thus −2 > −7, even though 7 is larger than 2 when the signs are removed.

The same rule applies to fractions and decimals. The number −0.2 is greater than −0.7 because it lies closer to zero. A temperature of −0.2 degrees is also warmer than −0.7 degrees on the same scale. Reading the context can help confirm the numerical order.

A rational number can be expressed as a ratio of integers with a nonzero denominator. Negative fractions, negative integers, terminating decimals, and repeating decimals can all be rational. Their different written forms do not create different ordering rules.

Value and absolute value answer different questions

The absolute value of a number is its distance from zero. It is never negative. For example, |−8| = 8 and |−3| = 3. Although −8 < −3, the absolute value of −8 is greater because it is farther from zero.

A question asking for the greatest number wants numerical value. A question asking for the greatest magnitude, largest absolute value, or greatest distance from zero wants something else. If the choices are −0.9, −0.4, and −0.1, the greatest value is −0.1, while the greatest absolute value belongs to −0.9.

This distinction is useful when interpreting signed quantities. A larger debt magnitude can be represented by a smaller signed account balance. A larger drop can be represented by a more negative change. Translate words such as largest into the specific measurement being compared.

Compare negative decimals by place value

Consider −0.62 and −0.607. First compare the positive magnitudes 0.620 and 0.607. At the tenths place they agree; at the hundredths place, 2 is greater than 0. Therefore 0.620 > 0.607. Restoring the negative signs reverses the order: −0.62 < −0.607.

Appending zeros to the right of a decimal does not change its value. Writing −0.62 as −0.620 simply aligns the place values. Counting decimal digits alone would be misleading: more digits do not necessarily mean a larger or smaller number.

For −1.03 and −1.3, write the second as −1.30. The magnitude 1.03 is smaller than 1.30, so −1.03 is greater. Both numbers are negative, but comparing their magnitudes still requires ordinary decimal place value before the final reversal.

Fractions with a common positive denominator

Compare −5/12 and −7/12. Both use twelfths as the unit. Five negative twelfths lie closer to zero than seven negative twelfths, so −5/12 > −7/12. Once the denominator is the same and positive, compare the signed numerators directly.

If the denominators differ, find equivalent fractions. For −3/4 and −5/6, a common denominator is 12: −3/4 = −9/12 and −5/6 = −10/12. Since −9 > −10, the first fraction is greater.

Be careful with a negative denominator. The fraction 3/(−4) equals −3/4, and (−3)/(−4) equals positive 3/4. Move signs into a consistent position before comparing. A fraction with two negative signs may belong on the positive side of zero, where it is greater than every negative number.

Fractions and decimals together

Compare −7/8 with −0.86. Because 7/8 = 0.875 exactly, the comparison becomes −0.875 versus −0.860. The first lies farther left, so −7/8 < −0.86. Three decimal places make the decision visible without rounding.

For −2/3 versus −0.67, the decimal −0.6666... is greater than −0.6700.... Rounding −2/3 to −0.67 too early would erase a real difference. When two values are close, keep enough precision to distinguish them or convert the terminating decimal into a fraction.

Using fractions, −0.67 = −67/100. Compare −2/3 and −67/100 with denominator 300: they become −200/300 and −201/300. The first is greater by 1/300. This exact approach avoids any dependence on a calculator's displayed decimal length.

Cross multiplication with signs under control

If b and d are positive, comparing a/b and c/d is equivalent to comparing ad and bc. Multiplying both fractions by the positive quantity bd preserves their order. For −4/7 and −5/9, compare −4 × 9 = −36 with −5 × 7 = −35. Because −36 < −35, −4/7 < −5/9.

The positive-denominator condition matters. Multiplying an inequality by a negative quantity reverses it. Normalize the fraction signs first so the shortcut has one clear rule. This avoids accidentally changing the comparison twice or not at all.

You can also compare the positive magnitudes 4/7 and 5/9, then reverse the result. Their cross products are 36 and 35, so 4/7 > 5/9 and consequently −4/7 < −5/9. Either method works if the signs are handled consistently.

Put a mixed set in increasing order

Order −0.72, −3/4, 0, −2/3, and 1/5 from least to greatest. First separate negatives, zero, and positives. Every negative belongs before zero, and the positive fraction belongs after it. Then compare the three negatives.

Their positive magnitudes are 0.72, 0.75, and 0.6666.... From largest magnitude to smallest, they are 0.75, 0.72, and 0.6666.... The negative values therefore increase in the order −3/4, −0.72, −2/3. The complete increasing list is −3/4, −0.72, −2/3, 0, 1/5.

For decreasing order, reverse that entire sequence. Avoid reversing only the negative portion while leaving zero and positive values in their earlier positions. Read whether the question asks for ascending, descending, least, or greatest before selecting the final option.

Interpret negative mixed numbers correctly

The conventional mixed number −2 1/4 means −(2 + 1/4), which is −2.25. It does not mean −2 + 1/4, which would equal −1.75. Parentheses make the intended grouping clearer when converting to an improper fraction: −2 1/4 = −9/4.

Thus −2 1/4 is less than −2 1/5 because −2.25 < −2.2. Comparing only the denominators or ignoring the whole-number part could lead to the wrong answer. Convert the complete signed mixed number, then apply the same number-line rule.

If the prompt writes an explicit expression such as −2 + 1/4, follow its addition sign. The written mixed-number convention and an algebraic sum have different meanings. Accurate reading comes before arithmetic.

Check equality before forcing an order

Some choices are equivalent forms of the same number. For example, −0.5, −1/2, and −4/8 are equal. Simplification or conversion can reveal that no strict inequality exists between them. Use an equals sign if the task permits it, or place equivalent values together when ordering a list.

Zero also deserves attention. The expressions −0 and 0 denote the same real number. A very small negative decimal is less than zero even if a rounded display shows zero. Use the value supplied in the problem rather than introducing rounding that changes the comparison.

A reliable final check

After comparing, place the results mentally from left to right. The more negative values must be farther left, zero separates negative and positive numbers, and equal values occupy the same location. For close fractions, confirm with a common denominator or a signed cross product.

The central question is which value lies farther right. Absolute value, digit count, numerator size, and denominator size are useful only when interpreted within the full signed number. Keeping that relationship visible makes unfamiliar combinations of fractions and decimals manageable.