The Big Idea
To compare numbers is to determine their relationship: less than, equal to, or greater than. To order numbers is to arrange several values according to those comparisons.
The real number line gives the central rule:
\[ a<b \quad\Longleftrightarrow\quad a\text{ lies to the left of }b. \]
| Relation | Meaning | Number-line interpretation |
|---|---|---|
| \(a<b\) | \(a\) is less than \(b\) | \(a\) lies left of \(b\) |
| \(a=b\) | \(a\) and \(b\) have the same value | They identify the same point |
| \(a>b\) | \(a\) is greater than \(b\) | \(a\) lies right of \(b\) |
Comparison Notation
| Symbol | Read as | Example |
|---|---|---|
| \(=\) | is equal to | \(\dfrac12=0.5\) |
| \(\ne\) | is not equal to | \(\sqrt2\ne1.4\) |
| \(<\) | is less than | \(-4<3\) |
| \(>\) | is greater than | \(7>5\) |
| \(\le\) | is less than or equal to | \(x\le8\) |
| \(\ge\) | is greater than or equal to | \(y\ge0\) |
The pointed end of \(<\) or \(>\) faces the smaller value; the open side faces the larger value. More importantly, read the entire statement aloud rather than relying only on the symbol’s shape.
The Trichotomy Property
For any real numbers \(a\) and \(b\), exactly one of these statements is true:
\[ a<b,\qquad a=b,\qquad a>b. \]
This is the trichotomy property. Real numbers are totally ordered: every pair can be compared.
Not every mathematical object has this kind of natural order. Complex numbers, matrices, and vectors may be compared by selected features, but they do not inherit one universal real-number-style \(<\) relation.
Different Forms Can Name the Same Point
Before deciding that one representation is larger, determine whether the forms are equivalent:
\[ \frac12=0.5=50\%=2^{-1}. \]
These expressions look different but identify the same real number. Likewise,
\[ 0.999\ldots=1. \]
Comparison concerns value, not the length or appearance of the notation.
Compare Signs First
Sign often settles a comparison immediately:
- Every positive real number is greater than \(0\).
- Every negative real number is less than \(0\).
- Every positive real number is greater than every negative real number.
\[ -12<0<0.001. \]
Magnitude becomes relevant after the signs are understood.
Comparing Negative Numbers
Among negative numbers, the value closer to zero is greater:
\[ -8<-3. \]
Although \(8>3\), negation reflects both values across the origin and reverses their order:
\[ 8>3 \quad\Longrightarrow\quad -8<-3. \]
A larger absolute value means farther from zero, not automatically a larger signed number.
Value and Magnitude Are Different Comparisons
Absolute value compares distance from zero:
\[ |-9|=9, \qquad |4|=4. \]
Therefore \(|-9|>|4|\), even though \(-9<4\).
| Question | Comparison | Reason |
|---|---|---|
| Which value is greater? | \(4>-9\) | \(4\) lies farther right |
| Which magnitude is greater? | \(|-9|>|4|\) | \(-9\) lies farther from zero |
Ordering Integers
To arrange integers in ascending order, place the most negative values first, then zero, then the positive values from smaller to larger.
For the collection
\[ 6,\ -2,\ 0,\ -9,\ 4, \]
ascending order is
\[ -9<-2<0<4<6. \]
Descending order reverses the list:
\[ 6>4>0>-2>-9. \]
Comparing Decimals by Place Value
Align decimal points and compare corresponding place values from left to right. The first unequal place determines the order.
Compare \(4.307\) and \(4.32\). Write a trailing zero:
\[ 4.307 \quad\text{and}\quad 4.320. \]
The ones and tenths match. At the hundredths place, \(0<2\), so
\[ 4.307<4.32. \]
Appending zeros to the right of a terminating decimal does not change its value: \(4.32=4.320=4.3200\).
Negative Decimals Need a Two-Stage Check
First compare their positive magnitudes, then reverse the order because both values are negative.
Since
\[ 2.81>2.79, \]
it follows that
\[ -2.81<-2.79. \]
The number \(-2.79\) lies closer to zero and is therefore greater.
Compare Fractions with a Common Denominator
When fractions have the same positive denominator, compare their numerators:
\[ \frac38<\frac58. \]
With different denominators, rewrite the fractions using a common positive denominator:
\[ \frac34=\frac{15}{20}, \qquad \frac45=\frac{16}{20}. \]
Therefore
\[ \frac34<\frac45. \]
Cross-Product Comparison
For positive denominators \(b\) and \(d\),
\[ \frac ab<\frac cd \quad\Longleftrightarrow\quad ad<bc. \]
Compare \(5/7\) and \(7/10\):
\[ 5\cdot10=50, \qquad 7\cdot7=49. \]
Since \(50>49\),
\[ \frac57>\frac7{10}. \]
Positive denominators are essential. If a denominator is negative, normalize the sign first; multiplying an inequality by a negative quantity reverses its direction.
Use Fraction Benchmarks
Benchmarks such as \(0\), \(1/2\), and \(1\) can settle a comparison quickly.
Compare \(4/9\) and \(5/8\):
\[ \frac49<\frac12 \quad\text{because}\quad 2(4)<9, \]
\[ \frac58>\frac12 \quad\text{because}\quad 2(5)>8. \]
Therefore \(4/9<5/8\) without finding a common denominator.
Mixed Numbers and Improper Fractions
Compare whole-number parts first. If they match, compare the fractional parts:
\[ 3\frac25<3\frac12 \]
because \(2/5<1/2\).
Converting to improper fractions or decimals is also valid:
\[ 3\frac25=\frac{17}{5}=3.4. \]
Choose the representation that makes the comparison most transparent.
Percentages Must Share a Common Interpretation
A percentage is a ratio per hundred:
\[ 37\%=0.37=\frac{37}{100}. \]
Thus \(42\%>0.4\) because \(0.42>0.40\).
In applications, compare percentages only when they describe compatible quantities. A \(10\%\) tax rate and a \(10\%\) probability have the same numerical value but different meanings and units of interpretation.
The Difference Test
To compare \(a\) and \(b\), examine the sign of their difference:
\[ \begin{aligned} a-b<0 &\Longrightarrow a<b,\\ a-b=0 &\Longrightarrow a=b,\\ a-b>0 &\Longrightarrow a>b. \end{aligned} \]
Compare \(7/12\) and \(4/7\):
\[ \frac7{12}-\frac47 =\frac{49-48}{84} =\frac1{84}>0. \]
Therefore \(7/12>4/7\).
The Ratio Test for Positive Values
When \(a>0\) and \(b>0\), the ratio \(a/b\) provides another comparison:
\[ \frac ab<1\Longrightarrow a<b, \qquad \frac ab>1\Longrightarrow a>b. \]
This method is especially useful for multiplicative comparisons, growth factors, rates, and relative size. It should not be applied without modification when signs or zero are involved.
Comparing Radicals
Nonnegative square roots preserve order:
\[ 0\le a<b \quad\Longrightarrow\quad \sqrt a<\sqrt b. \]
To compare \(\sqrt7\) and \(5/2\), square both nonnegative values:
\[ (\sqrt7)^2=7, \qquad \left(\frac52\right)^2=\frac{25}{4}=6.25. \]
Since \(7>6.25\), we have
\[ \sqrt7>\frac52. \]
Squaring preserves this comparison because both original quantities are known to be nonnegative.
Bound an Irrational Number
Locate a square root between nearby perfect squares:
\[ 25<30<36 \quad\Longrightarrow\quad 5<\sqrt{30}<6. \]
Decimal refinement can narrow the interval:
\[ 5.47^2=29.9209, \qquad 5.48^2=30.0304, \]
so \(5.47<\sqrt{30}<5.48\).
Comparing Powers
When positive powers share a base greater than \(1\), the larger exponent gives the larger value:
\[ 2^5<2^7. \]
For a base between \(0\) and \(1\), the order reverses as the exponent increases:
\[ \left(\frac12\right)^5 > \left(\frac12\right)^7. \]
Negative bases introduce parity and domain issues. Compare the actual structure rather than assuming that a larger exponent always gives a larger value.
Comparing Scientific Notation
A positive number in normalized scientific notation has the form
\[ a\times10^n, \qquad 1\le a<10. \]
Compare exponents first. If the exponents match, compare coefficients:
\[ 7.2\times10^6 > 9.8\times10^5, \]
\[ 4.3\times10^{-4} < 6.1\times10^{-4}. \]
For negative numbers, compare magnitudes and then reverse the order.
Exact and Approximate Comparisons
An exact value and a rounded approximation are not generally equal:
\[ \pi\approx3.14, \qquad \pi\ne3.14. \]
Since \(\pi\approx3.14159\), we may state
\[ 3.14<\pi<3.15. \]
When values are close, retain enough digits to support the comparison. Premature rounding can make unequal quantities appear equal or reverse a decision near a boundary.
Rounding Creates Intervals of Possible Values
A measurement reported as \(6.2\) to the nearest tenth commonly represents values from \(6.15\) up to \(6.25\), with endpoint treatment determined by the rounding convention.
Two rounded reports may match even when the underlying values differ. Conversely, overlapping uncertainty intervals may prevent a confident greater-than claim.
Distinguish these statements:
- the displayed numbers are equal;
- the underlying measurements are known to be equal;
- the available precision cannot distinguish them.
Convert Compatible Units Before Comparing
Numerical values with units must be expressed on a common scale:
\[ 2.5\text{ m}=250\text{ cm}. \]
Therefore \(2.5\text{ m}>240\text{ cm}\).
Quantities of different dimensions are not directly ordered by magnitude. Asking whether \(3\text{ kg}\) is greater than \(5\text{ s}\) is not a meaningful physical comparison without an additional model connecting mass and time.
Rates Require a Common Basis
Suppose one package costs \(\$8\) for \(5\) units and another costs \(\$11\) for \(8\) units. Compare unit prices:
\[ \frac{\$8}{5}=\$1.60\text{ per unit}, \qquad \frac{\$11}{8}=\$1.375\text{ per unit}. \]
The second package has the lower unit price. Comparing total prices alone would answer a different question.
A valid comparison aligns the quantity, unit, and basis of measurement.
Signed Numbers in Context
Negative numbers may represent temperature below a reference, debt, elevation below sea level, reverse motion, or net loss.
A balance of \(-\$20\) is greater than a balance of \(-\$50\):
\[ -20>-50. \]
Yet the magnitude of the second debt is greater:
\[ |-50|>|-20|. \]
Context determines whether we care about signed value, magnitude, gain, loss, distance, or another derived quantity.
How Operations Affect Order
| Transformation | Result | Effect |
|---|---|---|
| Add \(c\) | \(a+c<b+c\) | Order preserved |
| Multiply by \(c>0\) | \(ac<bc\) | Order preserved |
| Multiply by \(c<0\) | \(ac>bc\) | Order reversed |
| Apply an increasing function \(f\) | \(f(a)<f(b)\) | Order preserved on the interval |
| Apply a decreasing function \(f\) | \(f(a)>f(b)\) | Order reversed on the interval |
Building an Ordered Chain
Convert difficult values into comparable forms, then place them in a single ascending or descending chain.
Order
\[ -\frac32,\quad -1.4,\quad 0,\quad \sqrt2,\quad \frac75. \]
Useful comparisons are
\[ -\frac32=-1.5,\qquad \frac75=1.4,\qquad \sqrt2\approx1.414. \]
Therefore
\[ -\frac32<-1.4<0<\frac75<\sqrt2. \]
Comparison Determines Interval Membership
A number belongs to an interval when it satisfies the interval’s endpoint comparisons.
The value \(x=3\) belongs to \((1,5]\) because
\[ 1<3\le5. \]
The value \(x=1\) does not belong because the left endpoint is excluded. The value \(x=5\) does belong because the right endpoint is included.
Ordering supports interval notation, inequalities, domains, ranges, tolerances, and decision thresholds.
A Reliable Comparison Routine
- Identify what is being compared. Distinguish value, magnitude, distance, rate, and measured quantity.
- Check signs and zero. This may settle the comparison immediately.
- Normalize representation. Use common units, denominators, decimal places, bases, or powers.
- Choose an efficient method. Use number-line position, benchmarks, cross-products, a difference, a ratio, bounds, or a monotonic transformation.
- Preserve exactness when possible. Approximate only as precisely as the decision requires.
- Write the relation in words and symbols.
- Check reasonableness. Verify sign, scale, units, and order.
Common Confusions
Does the number with more digits have the greater value?
No. Place value, sign, scale, and representation determine value. For example, \(0.125<0.9\).
Is the negative number with the larger magnitude greater?
No. Among negative numbers, greater magnitude means farther left and therefore smaller.
Does adding trailing zeros change a terminating decimal?
Not to the right of the final decimal digit: \(2.4=2.40=2.400\).
Can fractions always be cross-multiplied without checking signs?
No. The standard comparison shortcut assumes positive denominators. Normalize denominator signs first.
Can I square both sides to compare any two numbers?
Squaring preserves order only on a domain where it is increasing, such as nonnegative real numbers. It can reverse or erase information when signs differ.
Is an approximation equal to the exact number?
Not generally. Use \(\approx\) for a rounded value unless the decimal representation is exact.
Can quantities with different units be compared directly?
Compatible units should first be converted to a common scale. Different physical dimensions need an additional model rather than direct numerical comparison.
Does a larger scientific-notation coefficient always mean a larger value?
No. Compare powers of ten first when the signs match; compare coefficients only when the exponents match.
Do equal rounded measurements prove equal underlying values?
No. Rounding may map a range of underlying values to the same displayed number.
Can every mathematical object be ordered like a real number?
No. Real numbers have a total order, but complex numbers, vectors, and matrices require selected comparison criteria.
Working Summary
Comparison determines whether one value is less than, equal to, or greater than another; ordering extends those comparisons to a list. On the standard number line, farther left means smaller and farther right means larger.
Signs should be compared before magnitudes. Fractions, decimals, percentages, radicals, powers, and scientific notation each offer efficient comparison methods, but equivalent forms and domain conditions must be recognized first.
Meaningful applied comparisons require compatible units, common rates, and appropriate precision. A strong comparison states the method, preserves exactness as long as useful, and checks sign, scale, and context.
Check Your Understanding
Compare \(-12\) and \(-7\).
\(-12<-7\) because \(-12\) lies farther left.
Compare \(3.407\) and \(3.47\).
Write \(3.47\) as \(3.470\). At the hundredths place, \(0<7\), so \(3.407<3.47\).
Compare \(5/6\) and \(7/9\).
Cross-products give \(5(9)=45\) and \(7(6)=42\). Therefore \(5/6>7/9\).
Compare \(-3/4\) and \(-0.8\).
\(-3/4=-0.75\), and \(-0.75>-0.8\).
Compare \(\sqrt{20}\) and \(9/2\).
Both are nonnegative. Their squares are \(20\) and \(81/4=20.25\), so \(\sqrt{20}<9/2\).
Compare \(3^8\) and \(3^6\).
The common base \(3>1\), so the larger exponent gives the larger value: \(3^8>3^6\).
Compare \((1/3)^4\) and \((1/3)^6\).
The base lies between \(0\) and \(1\), so the larger exponent gives the smaller value: \((1/3)^4>(1/3)^6\).
Order \(\pi\), \(3.14\), and \(22/7\).
\(3.14<\pi<22/7\). Numerically, \(\pi\approx3.14159\) and \(22/7\approx3.142857\).
Which is greater: \(2.4\text{ m}\) or \(235\text{ cm}\)?
\(2.4\text{ m}=240\text{ cm}\), so \(2.4\text{ m}>235\text{ cm}\).
Arrange \(-2\), \(0.25\), \(-1/3\), and \(0\) in ascending order.
\(-2<-1/3<0<0.25\).