Mathematics Year 7: Rational numbers on the number line: comparing and ordering
Mathematics, Year 7 (Portugal): Every rational number has its place on the number line, and the further right it is, the greater it is: that is how fractions, decimals and negatives are compared and ordered together.
Every number has its point
On the number line, 0 is the reference, the positive numbers go to the right and the negative ones to the left; the opposite of a number sits at the same distance from 0, on the other side. Every rational number has exactly one point. To place −7/4, write it as an integer plus a proper fraction: −7/4 = −2 + 1/4. It lies between −2 and −1, a quarter of a unit to the right of −2. The further right, the greater the number.
Why an integer plus a fraction
The integer tells you between which two marks the number falls; the proper fraction tells you how far to walk from that mark. So you never place a point by guessing. The line also puts fractions, decimals and negatives in the same place, so they can be compared directly. It explains the order rules: every positive number is greater than 0, every negative number is smaller than 0, and of two negatives the one closer to 0 is the greater, as with the integers.
Ordering five numbers
Order from least to greatest: −3/4, 0.5, −1.2, 5/4, −0.8. Step 1: write them all in the same form, decimal: −0.75; 0.5; −1.2; 1.25; −0.8. Step 2: negatives lie to the left of 0 and positives to the right, so −0.75, −1.2 and −0.8 come before 0.5 and 1.25. Step 3: among negatives, the one further from 0 is the smaller: the distances are 1.2 > 0.8 > 0.75, so −1.2 < −0.8 < −0.75. Result: −1.2 < −0.8 < −3/4 < 0.5 < 5/4.
Bigger digits, smaller number
Which is greater, −1/2 or −1/3? Many people notice that 1/2 is greater than 1/3 and answer −1/2. But on the number line −1/2 is further left than −1/3, so it is the smaller: −1/2 < −1/3. With a common denominator it shows: −3/6 < −2/6. Among negative numbers the order reverses: the greater the distance from 0, the smaller the number. Before you answer, place both on a line and ask which is more to the left.
Thermometers, depths and sorted lists
A thermometer is a number line standing upright: −2.5 °C is colder than −1.5 °C, because it is lower, even though 2.5 is greater than 1.5. A depth scale works the same way: −0.5 km is deeper, so lower, than −0.2 km. When you sort a column of negative decimals from smallest to greatest in a spreadsheet, the computer uses exactly this order: −1.5 comes before −0.25.
Test what you learned
Where is −7/4 on the number line?
- a) Between 1 and 2, because 7/4 lies between 1 and 2.
- b) Between −2 and −1, closer to −1.
- c) Between −2 and −1, closer to −2, because −7/4 = −2 + 1/4.
- d) Between −7 and −4, at the marks given by the numerator and the denominator.
Correct answer: c) Between −2 and −1, closer to −2, because −7/4 = −2 + 1/4. — Yes. −7/4 = −2 + 1/4 = −1.75: start at −2 and move a quarter of a unit to the right.
Which list puts 2/5, −3/5, −3/2 and −7/5 in increasing order, from least to greatest?
- a) −3/5 < −7/5 < −3/2 < 2/5
- b) −3/2 < −7/5 < −3/5 < 2/5
- c) 2/5 < −3/5 < −7/5 < −3/2
- d) −7/5 < −3/2 < −3/5 < 2/5
Correct answer: b) −3/2 < −7/5 < −3/5 < 2/5 — Yes: −1.5 < −1.4 < −0.6 < 0.4. The negatives come first, and the one furthest from 0 is the smallest.
Which is the greater number: −5/6 or −3/4?
- a) −3/4, because it is closer to 0 (−9/12 is to the right of −10/12).
- b) −5/6, because 5/6 is greater than 3/4.
- c) They are equal, because in both the denominator is 1 more than the numerator.
- d) It cannot be decided, because the denominators are different.
Correct answer: a) −3/4, because it is closer to 0 (−9/12 is to the right of −10/12). — Yes. With denominator 12, −5/6 = −10/12 and −3/4 = −9/12. On the line −9/12 is to the right of −10/12, so −3/4 is the greater.
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