Negative fractions and decimals on the number line
Some amounts on the far side of a zero are not whole, such as a find part of the way down a meter of soil, or a month with two and a half centimeters less snow than usual. Where do they sit on the number line?
Where you are 10 in Rational numbers
Why this exists
Negative numbers gave a name to each whole step left of zero: -1, -2, -3 and on. But the amounts that call for negative numbers are often not whole. A town’s snowfall in a month can be 2.4 centimeters below its average. A find in an archaeological dig can lie a quarter of a meter past the one-meter mark below the ground. Each is on the far side of a zero, and each falls between two integers.
On the right of zero this was settled earlier: fractions and decimals fill the gaps between the whole numbers. This lesson carries them across zero, so that every fraction and every decimal has a twin on the left. Together with the integers they make up the rational numbers, the numbers you can place on a line by cutting the unit into equal parts and counting.
Definition
Negative fractions. For a fraction such as \displaystyle \frac{3}{4}, the number \displaystyle -\frac{3}{4}, read “negative three quarters”, is the point \displaystyle \frac{3}{4} of a unit to the left of 0. To find it, split the stretch from 0 to -1 into 4 equal parts, as many as the denominator says, and count 3 of those parts, as many as the numerator says, leftward from 0. This is the same cutting and counting as in locating a fraction, done on the other side of zero.
Negative decimals. A decimal that stops is a fraction with denominator 10, 100, 1000 and so on, so it is placed the same way. -0.7 is seven tenths of a unit left of 0, and -0.07 is seven hundredths of a unit left of 0.
Negative mixed numbers. The minus sign in front of a mixed number belongs to the whole of it. \displaystyle -2\frac{1}{3} is \displaystyle 2\frac{1}{3} units left of 0: past -2 by another third of a unit, on the side away from 0, so between -3 and -2. An improper fraction such as \displaystyle -\frac{7}{4} is placed most easily as the mixed number with the same value, \displaystyle -1\frac{3}{4}; mixed numbers and improper fractions covers that rewriting.
Opposites of fractions and decimals. As with integers, the opposite of a number is the number the same distance from 0 on the other side. The opposite of \displaystyle \frac{2}{5} is \displaystyle -\frac{2}{5}, the opposite of -1.3 is 1.3, and \displaystyle -\left(-\frac{1}{2}\right), read “the opposite of negative one half”, is \displaystyle \frac{1}{2}.
Rational numbers. In this lesson a fraction has a whole number on top and a counting number underneath, such as \displaystyle \frac{3}{4}, \displaystyle \frac{7}{3} or \displaystyle \frac{0}{1}. A rational number is a fraction or the opposite of a fraction. Every integer is one: \displaystyle 5 = \frac{5}{1}, -5 is the opposite of \displaystyle \frac{5}{1}, and \displaystyle 0 = \frac{0}{1}. Every decimal that stops is one, since \displaystyle 1.3 = \frac{13}{10} and -0.07 is the opposite of \displaystyle \frac{7}{100}. So is every mixed number and its opposite, since a mixed number is a fraction written another way: \displaystyle 2\frac{1}{3} = \frac{7}{3}. On the line, the rational numbers are the points reached from 0 by cutting the unit into equal parts and counting some of them, right or left.
Intuition
Fold the number line at 0, like a page, and the left half lands on the right half. Every mark on the right meets a mark on the left: \displaystyle \frac{1}{2} meets \displaystyle -\frac{1}{2}, 0.7 meets -0.7, and \displaystyle 2\frac{1}{3} meets \displaystyle -2\frac{1}{3}. So nothing new has to be invented for the left side. Whatever cuts you made between 0 and 1 to place a fraction, make the same cuts between 0 and -1 and count the same number of pieces, only leftward.
Counting from zero always works, but it is often quicker to count from the nearest integer. To place -1.6, you can count 16 tenths left from 0, or go to -1 first and count 6 more tenths. Both land on the same point, because 1.6 is 1 and 6 tenths. On the left side every count runs away from 0, so the 6 tenths past -1 go toward -2.
Results
Every fraction and decimal has an opposite. Its opposite lies the same distance from 0 on the other side. A positive number’s opposite is negative, a negative number’s opposite is positive, and the opposite of 0 is 0.
The opposite of the opposite is the number you started with. For example,
A negative mixed number lies between two negative integers. \displaystyle -2\frac{1}{3} lies between -3 and -2, a third of a unit past -2 on the side away from 0.
Equal numbers have equal opposites. If two fractions or decimals name the same point, so do their opposites. Since \displaystyle \frac{1}{2} = 0.5 = \frac{2}{4}, also \displaystyle -\frac{1}{2} = -0.5 = -\frac{2}{4}.
What is rational. Every integer, every fraction, every mixed number and every decimal that stops is a rational number, and so is the opposite of each.
Why it is true
Opposites. Folding the line at 0 keeps every distance from 0 and swaps the two sides. A fraction placed by cutting the unit from 0 to 1 into equal parts and counting some of them is carried to the point reached by the same cuts and the same count between 0 and -1, which is exactly how its negative was placed. So a number and its negative are the same distance from 0 on opposite sides, which makes them opposites. Folding twice puts every point back where it started, which is why the opposite of the opposite is the number itself.
Mixed numbers. \displaystyle 2\frac{1}{3} lies past 2 and before 3, so the fold carries it past -2 and before -3.
Equal numbers, equal opposites. The fold carries one point on the right to one point on the left, so two names for one point have one opposite.
What is rational. A whole number is itself over 1, and a negative integer is the opposite of that. A decimal that stops counts tenths, hundredths or the like, so it is a fraction with denominator 10, 100 and so on, or the opposite of one. A mixed number equals an improper fraction.
Worked examples
Locating a negative decimal
Place -1.6 on a number line. Name it also as a mixed number and as a fraction.
SolutionThe number 1.6 is more than 1 and less than 2, so its opposite -1.6 lies between -2 and -1. The decimal counts tenths, so split the unit from -1 to -2 into 10 equal parts. Start at -1 and count 6 of those tenths further left, away from 0. That point is -1.6. Counting all the way from 0 gives the same point, 16 tenths left of 0.
As a mixed number, \displaystyle 1.6 = 1\frac{6}{10} = 1\frac{3}{5}, where \displaystyle \frac{6}{10} = \frac{3}{5} by equivalent fractions, dividing top and bottom by 2. As a fraction, 1.6 is 16 tenths, and \displaystyle \frac{16}{10} = \frac{8}{5}. The opposites of equal numbers are equal, so
A mixed number and an improper fraction
Place \displaystyle -2\frac{1}{3} and \displaystyle -\frac{7}{4} on one number line.
SolutionThe minus sign in \displaystyle -2\frac{1}{3} covers the whole mixed number, so the point is \displaystyle 2\frac{1}{3} units left of 0. That is 2 whole units, which reaches -2, then a third of a unit further left. So \displaystyle -2\frac{1}{3} lies between -3 and -2. Split the unit from -2 to -3 into thirds and take the first mark past -2.
For \displaystyle -\frac{7}{4}, find the whole units in \displaystyle \frac{7}{4} first. Four quarters make 1, and 3 quarters remain, so \displaystyle \frac{7}{4} = 1\frac{3}{4} and \displaystyle -\frac{7}{4} = -1\frac{3}{4}. It lies between -2 and -1. Split the unit from -1 to -2 into quarters and count 3 quarters left from -1.
Each fraction needed its own cuts, thirds in one unit and quarters in another. A line may be cut differently in different units, as long as the parts inside each unit are equal.
Opposites, and showing a number is rational
(a) Find the opposite of \displaystyle \frac{6}{11}. (b) Find the opposite of -3.2. (c) Simplify \displaystyle -\left(-2\frac{5}{6}\right). (d) Show that 12, -4, 0.09 and \displaystyle -1\frac{3}{8} are all rational numbers.
Solution(a) \displaystyle \frac{6}{11} is six elevenths of a unit right of 0. The point six elevenths left of 0 is \displaystyle -\frac{6}{11}, so the opposite of \displaystyle \frac{6}{11} is \displaystyle -\frac{6}{11}.
(b) -3.2 lies 3.2 units left of 0, so its opposite is 3.2, the same distance to the right.
(c) Read it as “the opposite of negative two and five sixths”. Negative two and five sixths is \displaystyle 2\frac{5}{6} units left of 0, and its opposite is the same distance to the right:
(d) A rational number is a fraction or the opposite of one, so write each in that form. \displaystyle 12 = \frac{12}{1}, a fraction. -4 is the opposite of \displaystyle \frac{4}{1}. 0.09 is nine hundredths, \displaystyle \frac{9}{100}. And \displaystyle 1\frac{3}{8} = \frac{11}{8}, since 1 is 8 eighths and 3 more eighths make 11, so \displaystyle -1\frac{3}{8} is the opposite of \displaystyle \frac{11}{8}. All four are rational.
Finding zero and the scale from two opposites
On a number line the points \displaystyle -\frac{2}{3} and \displaystyle \frac{2}{3} are marked, and evenly spaced tick marks between them make 8 equal spaces. Where is 0, what does one space stand for, and what number is at each tick mark from \displaystyle -\frac{2}{3} to 0?
SolutionThe two marked points are opposites, so they are the same distance from 0, one on each side. Zero is therefore exactly halfway between them. The 8 spaces split into 4 on each side, so 0 is 4 spaces from either end.
From 0 to \displaystyle \frac{2}{3} is 4 spaces. Two thirds is the same as four sixths, \displaystyle \frac{2}{3} = \frac{4}{6}, so 4 spaces make 4 sixths and each space is \displaystyle \frac{1}{6} of a unit.
Counting left from 0, the tick marks are \displaystyle -\frac{1}{6}, \displaystyle -\frac{2}{6}, \displaystyle -\frac{3}{6} and \displaystyle -\frac{4}{6}. In simplest form, from \displaystyle -\frac{2}{3} up to 0, they are
Check. The tick just left of 0 is \displaystyle -\frac{1}{6} and the tick just right of it is \displaystyle \frac{1}{6}, and these are opposites, as the fold at 0 says they must be.
Snowfall above and below a town’s average
Elin keeps a record of the snowfall in a made-up mountain town, comparing each month with the town’s average for that month. November had 3.6 centimeters less snow than average, December \displaystyle 1\frac{1}{4} centimeters more, January exactly the average, February 2.4 centimeters less and March 1.25 centimeters less.
(a) Write each month as a single number, taking the average as zero and more snow as positive.
(b) Place the five numbers on a number line.
(c) Which two months are opposites?
(d) The next winter, one month had as much more snow than average as this November had less. What number does Elin write for it?
Solution(a) Less than average is the negative side: November -3.6, December \displaystyle 1\frac{1}{4}, January 0, February -2.4 and March -1.25, all in centimeters.
(b) -3.6 is 6 tenths past -3, toward -4. -2.4 is 4 tenths past -2, toward -3. -1.25 is a quarter past -1, toward -2, since \displaystyle 0.25 = \frac{1}{4}. \displaystyle 1\frac{1}{4} is a quarter past 1, and January sits on the zero itself.
(c) December and March. They are written differently, but \displaystyle 1\frac{1}{4} = 1.25, so the two numbers are the same distance from 0 on opposite sides: one month had 1.25 centimeters more snow than average and the other 1.25 centimeters less.
(d) The opposite situation of “3.6 centimeters less than average” is “3.6 centimeters more than average”, so Elin writes the opposite of -3.6, which is 3.6.
Depths in an archaeological dig
Fynn and Hilde measure heights in a dig from ground level at the edge of the trench, in meters, with up as positive. They find a coin 0.3 meters below ground level, a pot rim 0.8 meters below, a bone \displaystyle 1\frac{1}{4} meters below and a hearth stone 1.7 meters below. A survey pole stands beside the trench, its top 0.8 meters above ground level.
(a) Write each height as a number.
(b) The trench wall has a mark every tenth of a meter. Between which two marks is the bone?
(c) Which find’s height is the opposite of the height of the pole top?
Solution(a) Ground level is 0 and below is negative: coin -0.3, pot rim -0.8, bone \displaystyle -1\frac{1}{4}, hearth stone -1.7, and pole top 0.8. The figure draws the line upright.
(b) Write the bone’s height as a decimal. Since \displaystyle \frac{1}{4} = 0.25, \displaystyle -1\frac{1}{4} = -1.25. The depth 1.25 is more than 1.2 and less than 1.3, exactly halfway, so the bone lies halfway between the marks at -1.2 and -1.3.
(c) The pole top is 0.8 meters above ground level and the pot rim 0.8 meters below, so the pot rim’s height, -0.8, is the opposite of the pole top’s height, 0.8.
Boundaries
The parts must be equal. Placing \displaystyle -\frac{3}{4} means cutting the unit into 4 equal parts. Marks that are not evenly spaced do not show quarters, and counting them lands in the wrong place.
The sign covers the whole mixed number. \displaystyle -2\frac{1}{3} is the opposite of \displaystyle 2\frac{1}{3}. It lies a third of a unit past -2 on the side away from 0, between -3 and -2.
Zero is whatever the situation says it is. In Example 5 zero is the town’s average, not “no snow”. A month at -1.25 still had snow, only 1.25 centimeters less than usual.
Decimals that do not stop. The decimals in this lesson stop after a few digits. Some fractions, such as \displaystyle \frac{1}{3}, have a decimal that never stops, 0.333\ldots, and \displaystyle -\frac{1}{3} is still a rational number, being the opposite of a fraction; terminating and repeating decimals shows how such decimals are written.
No denominator of zero. A unit cannot be cut into 0 equal parts, so a fraction never has 0 underneath.
Not every point is rational. Some points on the line are not rational numbers, even though between any two fractions lies another: no fraction, however fine the cuts, lands on them exactly. They are the irrational numbers, met much later.
Where this goes next
Now that fractions and decimals sit on both sides of zero, they can be compared and put in order, which is comparing and ordering rational numbers; the rule that further right is greater, from integers on a number line, carries over unchanged. The distance a number sits from 0, which it shares with its opposite, becomes absolute value.
Two such lines crossing at right angles make the coordinate plane. Later, adding and subtracting rational numbers and multiplying and dividing them calculate with these numbers, and what makes a number rational restates the definition as one integer over another.
Exercises
Placing points
For 1 to 5, say between which two integers the number lies, and how you would cut the line to place it.
- \displaystyle -\frac{4}{7}
- -2.3
- \displaystyle -3\frac{2}{5}
- \displaystyle -\frac{13}{5}
- -0.38
Reading a number line
Exercises 6 to 9 use this number line, marked in quarters.
- Name point A as a mixed number, as a decimal and as a fraction.
- Name point B in the same three ways.
- Name points C and D, each as a fraction or mixed number and as a decimal.
- Which marked point is the opposite of \displaystyle 1\frac{1}{2}? Where does the opposite of A lie?
Opposites
- Find the opposite of \displaystyle \frac{5}{12}.
- Find the opposite of -4.05.
- Simplify \displaystyle -\left(-\frac{5}{6}\right).
- Simplify -(-(-0.6)).
- Which of these are pairs of opposites? (a) \displaystyle \frac{1}{2} and -0.5 (b) -0.9 and \displaystyle \frac{9}{10} (c) -2.2 and \displaystyle 2\frac{1}{5} (d) \displaystyle \frac{2}{3} and -0.6
Rational numbers
- Show that each is a rational number by writing it as a fraction or the opposite of a fraction: -9, 0, 3.7, \displaystyle -1\frac{2}{9}.
- Do -0.04, \displaystyle -\frac{4}{100} and \displaystyle -\frac{1}{25} name the same point? Explain.
- Name the number halfway between -2 and -3 in three ways: as a decimal, as a mixed number and as a fraction.
In the world
- In Dax’s town, January had 1.8 centimeters less snow than its average, February \displaystyle \frac{3}{5} of a centimeter more, and March exactly the average. Write each month as a number, with the average as zero and more snow as positive.
- A month’s snowfall is recorded as \displaystyle -4\frac{3}{4} centimeters, measured from the town’s average with more snow as positive. Say in words what that means, and give the opposite situation and its number.
- Kira finds a bead 0.65 meters below ground level. Write its height as a number, with ground level as zero and up as positive, and name the two tenth-of-a-meter marks it lies between.
- A layer of ash runs from \displaystyle 1\frac{2}{5} meters below ground level down to \displaystyle 1\frac{9}{10} meters below. Write the heights of its top and bottom as decimals.
- The top of a marker stake is 0.55 meters above ground level. A find lies at the opposite height. What is its height, and where is the find?
Reasoning
- Explain why \displaystyle -3\frac{1}{4} lies to the left of -3 and not between -3 and -2.
- Is every integer a rational number? Explain.
- Is there a rational number that is its own opposite? Explain.
Answers
- Between -1 and 0. Cut the unit from 0 to -1 into 7 equal parts and count 4 of them left from 0.
- Between -3 and -2. Cut the unit from -2 to -3 into tenths and count 3 tenths left from -2.
- Between -4 and -3. Cut the unit from -3 to -4 into fifths and count 2 fifths left from -3.
- Between -3 and -2. Ten fifths make 2 and 3 fifths remain, so \displaystyle -\frac{13}{5} = -2\frac{3}{5}. Cut the unit from -2 to -3 into fifths and count 3 fifths left from -2.
- Between -1 and 0. Cut that unit into hundredths and count 38 left from 0; or cut it into tenths first, and -0.38 lies 8 hundredths past the mark at -0.3, toward -0.4.
- \displaystyle A = -2\frac{1}{4} = -2.25 = -\frac{9}{4}. It is one quarter past -2, on the side away from 0, and \displaystyle 2\frac{1}{4} is 9 quarters.
- \displaystyle B = -1\frac{1}{2} = -1.5 = -\frac{3}{2}. It is two quarters, that is one half, past -1.
- \displaystyle C = -\frac{1}{4} = -0.25, one quarter left of 0. \displaystyle D = 2\frac{3}{4} = 2.75, three quarters right of 2.
- B, since \displaystyle B = -1\frac{1}{2}. The opposite of A is \displaystyle 2\frac{1}{4}, a quarter right of 2, which is two quarter marks left of D.
- \displaystyle -\frac{5}{12}. It is the same distance from 0, on the other side.
- 4.05. It is 4.05 units right of 0, as -4.05 is 4.05 units left.
- \displaystyle \frac{5}{6}. The opposite of negative five sixths is five sixths.
- -0.6. Work from the inside: -(-0.6) = 0.6, and the opposite of 0.6 is -0.6.
- (a), (b) and (c) are pairs of opposites, since \displaystyle \frac{1}{2} = 0.5, \displaystyle \frac{9}{10} = 0.9 and \displaystyle 2\frac{1}{5} = 2.2. (d) is not: \displaystyle 0.6 = \frac{3}{5} = \frac{9}{15} while \displaystyle \frac{2}{3} = \frac{10}{15}, so \displaystyle \frac{2}{3} and 0.6 are not the same distance from 0.
- -9 is the opposite of \displaystyle \frac{9}{1}; \displaystyle 0 = \frac{0}{1}; \displaystyle 3.7 = \frac{37}{10}; and \displaystyle -1\frac{2}{9} is the opposite of \displaystyle \frac{11}{9}, since \displaystyle 1\frac{2}{9} = \frac{11}{9}.
- Yes. 0.04 is four hundredths, \displaystyle \frac{4}{100}, and \displaystyle \frac{4}{100} = \frac{1}{25} by dividing top and bottom by 4. The three positive numbers are one point, so their opposites are one point too.
- \displaystyle -2.5 = -2\frac{1}{2} = -\frac{5}{2}. Halfway is half a unit past -2, away from 0.
- January -1.8, February \displaystyle \frac{3}{5}, March 0, all in centimeters.
- The month had \displaystyle 4\frac{3}{4} centimeters less snow than average. The opposite situation is a month with \displaystyle 4\frac{3}{4} centimeters more snow than average, which is \displaystyle 4\frac{3}{4}.
- -0.65 meters. It lies halfway between the marks at -0.6 and -0.7.
- Top -1.4 meters, bottom -1.9 meters, since \displaystyle \frac{2}{5} = \frac{4}{10} = 0.4 and \displaystyle \frac{9}{10} = 0.9.
- -0.55 meters: the find is 0.55 meters below ground level.
- \displaystyle -3\frac{1}{4} is the opposite of \displaystyle 3\frac{1}{4}, which lies a quarter past 3, between 3 and 4. The fold at 0 carries it to a quarter past -3 on the side away from 0, between -4 and -3. The numbers between -3 and -2 are the opposites of numbers between 2 and 3, and \displaystyle 3\frac{1}{4} is not one of those.
- Yes. A whole number such as 7 is \displaystyle \frac{7}{1}, a fraction, so it is rational, and \displaystyle 0 = \frac{0}{1}. A negative integer such as -7 is the opposite of \displaystyle \frac{7}{1}, which makes it rational too.
- Yes, exactly one: 0, whose opposite is 0 itself. Any other rational number lies on one side of 0 and its opposite on the other side, so the two cannot be the same point.