- Mark points with whole and fractional coordinates on the number line and read the coordinate of a point
- Decide whether a number is an integer, rational, positive or negative, and describe real situations with signed numbers
- Find the opposite and the absolute value of a number and solve simple equations |x| = a
- Compare positive and negative numbers and put them in order
A winter morning. The forecast says +6 °C in Baku and −8 °C at Shahdag. The −1 button in the lift takes you to the underground car park, and the banking app shows a card balance of −15 manat. In each of these numbers the sign carries important information: above or below zero, above or below ground, money you have or money you owe. How many degrees apart are +6 °C and −8 °C? In this lesson we answer that by counting units on the number line.
You already know natural numbers, common fractions and decimals (see the lessons “Common fractions: meaning, simplifying and comparing” and “Decimal fractions”). All of them are greater than or equal to zero. Now we extend the world of numbers to the left of zero. Calculating with negative numbers comes in the next lesson, “Operations with positive and negative numbers”.
The number line
A straight line on which three things are chosen: the origin O (the number 0 belongs to it), the positive direction (usually to the right, shown by an arrow) and the unit length (a segment of length 1). The number that belongs to a point is its coordinate: A(3) means “the coordinate of point A is 3”.
Positive numbers lie to the right of O and negative numbers to the left: 3 is the point 3 units to the right of O, and −3 is the point 3 units to the left. Fractions have their places too: −1.5 is exactly halfway between −1 and −2, and to find −2/3 you split the segment from O to −1 into three equal parts and count two of them to the left of O. A thermometer is a number line as well, just vertical: “+” above zero, “−” below.
- 1Look at the sign
If a > 0 you go to the right of O, if a < 0 to the left. If a = 0, the point is O itself.
- 2Count the units
Ignoring the sign, measure off that many unit lengths: for −4, 4 units.
- 3Split for the fraction
If the number is a fraction, split the unit into as many equal parts as the denominator: for −2.5, 2 whole units and half a unit; for −3/4, 3 of the 4 quarters of a unit.
- 4Label it
Mark the point and write its letter and coordinate: C(−2.5).
1) How do you mark the points A(4), B(−3), C(−1.5) and D(−3/4) on the number line?
2) Point M is 5 units to the left of O, and point N is 7 units to the right of M. Find their coordinates.
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2) M is on the left, so its coordinate is negative: M(−5). From M, 7 steps to the right: 5 steps bring us to O, and 2 more steps take us to the point 2. So N(2).
It is +6 °C in Baku and −8 °C at Shahdag. How many degrees apart are these temperatures? One mark on the thermometer is 1 °C.
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The points are on different sides of zero, so we add the marks: 8 + 6 = 14 °C.
For comparison: −8 °C and −3 °C are on the same side of zero, 8 − 3 = 5 marks apart, a difference of 5 °C.
Integers and rational numbers
Numbers greater than zero are positive, and numbers less than zero are negative. Zero is neither positive nor negative — it is the border between the two sides. The “+” in front of a positive number may be left out: +6 = 6. The “−” in front of a negative number is never left out: −6 and 6 are completely different numbers.
The natural numbers (1, 2, 3, …), their negative opposites (−1, −2, −3, …) and 0: Z = {…, −3, −2, −1, 0, 1, 2, 3, …}. On the number line the integers are the points one unit apart.
Numbers that can be written as m/n, where m is an integer and n is a natural number. They include all integers (5 = 5/1, −2 = −2/1), positive and negative fractions (−3/4) and terminating decimals (−0.25 = −1/4).
- ra rational number
- mthe numerator — any integer (positive, negative or 0)
- nthe denominator — a natural number (n ≠ 0)
Every natural number is an integer, and every integer is a rational number: N ⊂ Z ⊂ Q. The sign of a negative fraction can be written in the numerator: −3/4 = (−3)/4.
Numbers: 7; −4; 0; −2/3; 3.5; −0.125; 12/4; −6/3.
1) Which of them are integers?
2) Write each one in the form m/n to show that it is rational.
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2) 7 = 7/1; −4 = −4/1; 0 = 0/1; −2/3 is already in the form m/n; 3.5 = 35/10 = 7/2; −0.125 = −125/1000 = −1/8; 12/4 = 3/1; −6/3 = −2/1.
Conclusion: all eight numbers are rational, and five of them are integers.
In real life, negative numbers show the two opposite directions of a quantity. Which direction counts as positive is a matter of agreement, but once chosen it must be kept:
- Temperature: 5 degrees below zero is written −5 °C, 12 degrees above zero +12 °C.
- Height and depth: a diver 40 m below sea level is at −40 m, and the summit of Bazarduzu, 4466 m high, is at +4466 m.
- Bank account: 50 manat paid in is written +50 and 20 manat spent −20; a balance of −15 manat means a debt of 15 manat.
- Lift: the −2 button takes you to the second floor below ground.
Opposite numbers
Two numbers that differ only in their sign: 5 and −5, −2.7 and 2.7. On the number line they lie at the same distance from O on opposite sides, that is, they are symmetric about O. The opposite of 0 is 0 itself. The opposite of a number a is written −a.
Careful: read −a as “the opposite of a”, not as “a negative number”. If a = 4, then −a = −4 is negative; if a = −4, then −a = −(−4) = 4 is positive. Taking the opposite twice brings you back to where you started — like being reflected in a mirror twice.
- aany rational number
- −athe opposite of a (positive when a is negative)
The opposite of the opposite is the number itself. Two “−” signs next to each other cancel: −(−(−5)) = −5, because of three signs two cancel and one is left.
1) Write the opposites of 7; −3.5; 2/9; 0.
2) Simplify: −(−12); −(−(−8)); −(+6).
3) If −x = 9, what is x? If −y = −2.4, what is y?
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2) −(−12) = 12; −(−(−8)) = −8 (three “−”: two cancel, one is left); −(+6) = −6.
3) If the opposite of x is 9, then x is the opposite of 9: x = −9. If the opposite of y is −2.4, then y = 2.4.
Absolute value
The distance from the point that shows a number on the number line to the origin O. The absolute value of a is written |a| and read “the absolute value of a”. Since it is a distance, it is never negative.
- aany rational number
- |a|the absolute value of a — the distance from a to O
- −athe opposite of a; positive when a < 0
The absolute value of a positive number or of zero is the number itself; the absolute value of a negative number is its opposite. Therefore |a| ≥ 0 and |−a| = |a|: opposite numbers have equal absolute values.
At first sight the formula looks strange: an absolute value cannot be negative, so why “−a”? Because when a is negative, −a is positive. For example, if a = −7, then |a| = −a = −(−7) = 7. So here “−a” is not a negative number but the opposite number.
1) |−12|; |3.7|; |0|; |−2/5|.
2) |−8| + |3|; |−8| − |−3|; |−6| · |−2|; |−20| ÷ |4|.
3) Find the value of −|−9|.
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2) First find the absolute values, then calculate with ordinary positive numbers: 8 + 3 = 11; 8 − 3 = 5; 6 · 2 = 12; 20 ÷ 4 = 5.
3) First the absolute value: |−9| = 9, then its opposite: −|−9| = −9. The absolute value is not negative, but the “−” in front of it makes the result negative.
- xthe unknown number
- athe given distance — a positive number
There are two points at distance a from O: one on the right and one on the left. If |x| = 0, then only x = 0; an equation such as |x| = −3 has no solution, because a distance cannot be negative.
Solve the equations: 1) |x| = 5; 2) |x| = 2.5; 3) |x| = 0; 4) |x| = −4.
5) List all integers whose absolute value is less than 3.
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2) x = 2.5 or x = −2.5.
3) Only x = 0: the point at distance 0 from O is O itself.
4) No solution: an absolute value cannot be negative.
5) The integers less than 3 units from O: −2, −1, 0, 1, 2 — 5 numbers. −3 and 3 are not included: their absolute value equals 3, it is not less than 3.
Comparing and ordering numbers
On the number line numbers increase from left to right: the number further to the right is the greater one. Three rules follow. Every positive number is greater than zero. Every negative number is less than zero, and therefore less than any positive number. The most interesting rule is the third one — comparing two negative numbers.
- a, bthe negative numbers being compared
- |a|, |b|their distances from O
Of two negative numbers, the one with the larger absolute value is smaller: it lies further from zero, further to the left. A frost of −10 °C is colder than −2 °C, and a debt of 50 manat is worse than a debt of 5 manat.
Put <, > or = between the numbers: 1) −7 and −2; 2) −3.5 and −3.05; 3) −1/2 and −1/3; 4) −0.6 and −2/3; 5) |−4| and −5.
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2) 3.50 > 3.05, so −3.5 < −3.05.
3) 1/2 = 3/6 > 2/6 = 1/3, so −1/2 < −1/3.
4) 2/3 = 0.666… > 0.6, so −0.6 > −2/3.
5) |−4| = 4 is positive and −5 is negative: |−4| > −5.
- 1Sort into groups
Split the numbers into three groups: negatives, zero, positives. If there is an absolute value sign, work it out first: |−5| = 5 is positive.
- 2Order the negatives
Compare the negatives by their absolute values: the one with the largest absolute value is the smallest and goes first.
- 3Order the positives
Order the positive numbers as usual; if needed, bring fractions to a common denominator or turn them into decimals.
- 4Join and check
Negatives → 0 → positives. Check: does this order go from left to right on the number line?
1) Put the numbers in increasing order: 3; −4; 1/2; −2.5; 0; −1/4; |−6|.
2) Suppose that on a winter morning the thermometers show: Shahdag −8 °C, Gabala −1 °C, Baku +6 °C, Shaki −3 °C, Lankaran +9 °C. Find the coldest and the warmest place and write the temperatures in decreasing order.
3) How many integers lie between −3.5 and 2.4?
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Answer: −4 < −2.5 < −1/4 < 0 < 1/2 < 3 < |−6|.
2) The coldest place is Shahdag (−8 °C, the negative number with the largest absolute value), the warmest is Lankaran (+9 °C).
Decreasing order: 9 > 6 > −1 > −3 > −8.
3) The first integer to the right of −3.5 is −3, the last integer to the left of 2.4 is 2: −3, −2, −1, 0, 1, 2 — 6 integers.
- 1.|−9| =
- 2.−(−14) =
- 3.The opposite of −3.5:
- 4.−4 −9 (< or >)
- 5.The negative solution of |x| = 6: x =
- 6.The number of integers between −2 and 3:
You can now see positive and negative numbers on the number line, compare them and find their absolute values. In the next lesson, “Operations with positive and negative numbers”, you will learn to add, subtract, multiply and divide them — and the absolute value and the opposite number will be the main tools of every rule there.
Key points
- A number line has an origin, a positive direction and a unit length; positive numbers lie to the right of 0, negative numbers to the left, and 0 is neither positive nor negative.
- Integers are the natural numbers, their opposites and 0. Rational numbers can be written as m/n (m an integer, n a natural number); every integer is rational: N ⊂ Z ⊂ Q.
- Opposite numbers differ only in their sign and are symmetric about O; −(−a) = a, and −a is positive when a is negative.
- The absolute value is the distance to 0: |a| ≥ 0, |−a| = |a|. The equation |x| = a (a > 0) has two solutions, a and −a; |x| = −3 has none.
- The number further right is greater; of two negative numbers, the one with the larger absolute value is smaller: −7 < −2, −1/2 < −1/3.
Check yourself
12 questions. Every correct answer earns XP.