2.04a, 2.04b - Comparing numbers

2.04a, 2.04b - Comparing numbers

Which is larger: 38\frac38 or 38%38\%? Compare signed numbers on a number line, put fractions, decimals and percentages into a common form, and use equality and inequality symbols precisely.

Ordering signed numbers

Is 38\frac38 larger than 38%38\% because three eighths sounds close to four tenths? We will settle that comparison by converting both values. First, make sure the direction of number order is secure.

On a number line, numbers increase to the right. This remains true for decimal steps: −0.3-0.3 is greater than −0.8-0.8, because it lies further right. Its distance from zero is smaller, but its value is larger.

[DIAGRAM: asset_name: Ordering signed decimals; asset_slug: signed-decimal-order; caption: Values increase to the right; negative values closer to zero are greater.; description: A horizontal number line from negative one to positive one, with marked points at negative 0.8, negative 0.3, zero, positive 0.3 and positive 0.8. The points are labelled and a right-pointing arrow above the line is labelled increasing value.]
Values increase to the right

Ascending and descending order

Ascending order means smallest first. Descending order means largest first. Zero lies between negative and positive numbers and belongs to neither group.

For an ascending list, separate negatives, zero and positives first. Among negative values, the number with the greater distance from zero comes earlier.

Worked example 1: numbers on both sides of zero

Question. Write −1-1, 0.30.3, −0.8-0.8 and 00 in ascending order.

Plan. Order the negatives first, then zero, then the positive number.

−1<−0.8further left is smaller−0.8<0<0.3negative, zero, positive\begin{aligned} -1&<-0.8 &&\text{further left is smaller}\\ -0.8&<0<0.3 &&\text{negative, zero, positive} \end{aligned}

Answer. −1, −0.8, 0, 0.3-1,\ -0.8,\ 0,\ 0.3.

Check. Moving through the answer from left to right moves consistently right along the number line.

Change only the sign of 0.30.3. It now belongs with the negatives, but is still the nearest of those negatives to zero.

Worked example 2: one more negative value

Question. Write −1-1, −0.3-0.3, −0.8-0.8 and 00 in ascending order.

Plan. Put the three negative values in number-line order, then place zero last.

−1<−0.8compare distances below zero−0.8<−0.3<0continue towards zero\begin{aligned} -1&<-0.8 &&\text{compare distances below zero}\\ -0.8&<-0.3<0 &&\text{continue towards zero} \end{aligned}

Answer. −1, −0.8, −0.3, 0-1,\ -0.8,\ -0.3,\ 0.

Check. Adding 11 to every entry gives 0, 0.2, 0.7, 10,\ 0.2,\ 0.7,\ 1, which is still ascending. Moving every point the same distance preserves their order.

Use the decimal point as a fixed anchor. 0.09<0.10.09<0.1, because nine hundredths is less than ten hundredths; a longer decimal is not automatically larger.

Watch out

−0.8>−0.3-0.8>-0.3 is false. Eight tenths below zero is further left than three tenths below zero, so −0.8<−0.3-0.8<-0.3.

Keep zero in its correct position.

Comparing different number forms

Fractions, decimals and percentages are easiest to compare when they are written in the same form.

For 38\frac38 and 38%38\%, use 38=0.375\frac38=0.375 and 38%=0.3838\%=0.38. Write 0.380.38 as 0.3800.380 to align places. Compare from the left: units and tenths agree; the hundredths digit 77 is less than 88, so 38<38%\frac38<38\%. The opening question is settled.

You can instead use fractions with a common positive denominator. Keep enough digits when converting: prematurely rounding two values to the same decimal can hide their order.

Worked example 3: a common decimal form

Question. Write 38\frac38, 0.380.38 and 36%36\% in order of size, smallest first.

Plan. Convert to decimals and align thousandths, then return the original forms.

38=0.375divide or use thousandths0.38=0.380append a zero36%=0.360divide 36 by 100\begin{aligned} \frac38&=0.375 &&\text{divide or use thousandths}\\ 0.38&=0.380 &&\text{append a zero}\\ 36\%&=0.360 &&\text{divide 36 by 100} \end{aligned}

Thus 0.360<0.375<0.3800.360<0.375<0.380.

Answer. 36%, 38, 0.3836\%,\ \frac38,\ 0.38.

Check. In percentages, the same order is 36%<37.5%<38%36\%<37.5\%<38\%.

Change only the percentage from 36%36\% to 37.6%37.6\%. The values are closer, so the thousandths column now decides the comparison.

Worked example 4: close values

Question. Write 38\frac38, 0.380.38 and 37.6%37.6\% in order of size, smallest first.

Plan. Keep exact decimals to three places rather than rounding to two.

38=0.375exact conversion37.6%=0.376divide by 1000.38=0.380align places\begin{aligned} \frac38&=0.375 &&\text{exact conversion}\\ 37.6\%&=0.376 &&\text{divide by 100}\\ 0.38&=0.380 &&\text{align places} \end{aligned}

Thus 0.375<0.376<0.3800.375<0.376<0.380.

Answer. 38, 37.6%, 0.38\frac38,\ 37.6\%,\ 0.38.

Check. The gap from 0.3750.375 to 0.3760.376 is 0.0010.001; both are smaller than 0.3800.380.

For negative mixed forms, convert first and then respect number-line order. For instance, −34=−0.75<−0.7=−70%-\frac34=-0.75<-0.7=-70\%. Converting the forms does not change the rule for negatives. Values above one work too: 125%=1.25>1125\%=1.25>1.

Watch out

Do not round 0.3750.375 and 0.3760.376 to 0.380.38 before comparing them. Rounded equality does not mean the original values were equal.

Change only the percentage again and keep the original forms in your final list.

Equality and inequality symbols

A relation symbol states how two values compare; its precise meaning matters at equality.

SymbolMeaningExample
<<less than2<32<3
>>greater than3>23>2
≤\leless than or equal to2≤32\le3 and 3≤33\le3
≥\gegreater than or equal to3≥23\ge2 and 3≥33\ge3
==equal to0.4=0.400.4=0.40
≠\nenot equal to0.4≠0.040.4\ne0.04

Strict inequality

The symbols << and >> describe strict inequalities: they exclude equality. The symbols ≤\le and ≥\ge include equality. The open side of an inequality faces the greater value.

Different-looking forms may be equal. Compare values before selecting a symbol.

Worked example 5: two different negative values

Question. Put << or >> between −0.4-0.4 and −0.04-0.04.

Plan. Align hundredths and locate the values relative to zero.

−0.4=−0.40write hundredths−0.40<−0.04forty hundredths is further left\begin{aligned} -0.4&=-0.40 &&\text{write hundredths}\\ -0.40&<-0.04 &&\text{forty hundredths is further left} \end{aligned}

Answer. −0.4<−0.04-0.4<-0.04.

Check. Adding 0.40.4 to both sides gives 0<0.360<0.36, a true comparison.

Change only the right-hand value from −0.04-0.04 to −0.40-0.40. The values now coincide, so the strict comparison no longer applies.

Worked example 6: equality included

Question. Which of <<, >>, ≤\le, ≥\ge, ==, ≠\ne make a true statement between −0.4-0.4 and −0.40-0.40?

Plan. Establish equality, then select every symbol whose meaning includes it.

−0.4=−0.40a trailing zero changes nothing−0.4≤−0.40and−0.4≥−0.40equality is allowed\begin{aligned} -0.4&=-0.40 &&\text{a trailing zero changes nothing}\\ -0.4&\le-0.40\quad\text{and}\quad-0.4\ge-0.40 &&\text{equality is allowed} \end{aligned}

Answer. ==, ≤\le and ≥\ge.

Check. Neither value is strictly greater or smaller, and they are not unequal. This rules out the other three symbols.

Both a≤ba\le b and a≥ba\ge b can be true together: they force a=ba=b. This is a useful connection when you later describe intervals and solve inequalities.

Watch out

0.4≤0.40.4\le0.4 is true. “Less than or equal to” does not require the left side to be smaller; equality is enough.

Use the meanings of all six symbols, not their appearance alone.

Quick Recap

  • Values increase to the right on a number line, including below zero.
  • Convert mixed number forms to a common exact form before comparing.
  • Compare aligned decimal columns from the left and avoid premature rounding.
  • Strict inequalities exclude equality; ≤\le and ≥\ge include it.

38=0.375\frac38=0.375 is smaller than 38%=0.38038\%=0.380.