Upper and Lower Bounds. Intervals and Segments

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In this Math tutorial, you will learn:

  • What are the lower and upper bounds in numbers?
  • How to quickly express an estimated value of a mathematical expression using rounding?
  • Using the concepts of upper and lower bound in practice to find the minimum and maximum value of any quantity
  • What are intervals and segments? How are they expressed?
  • How to show intervals and segments in a number line?
  • What are half-intervals and half-segments? How are they expressed in symbols? How do we display them in a number line?
  • How do we show rounding on a number line?

Introduction

In many cases, it is worth knowing the range of numbers that provide a specific rounding. This is because we want to estimate how much we may err if giving an approximate value of a number. We have previously explained that in science we must speak with competence and not guess.

In this tutorial, we will explain how to identify the margins of error we accept when rounding a number. We call these allowed margins of error "bounds" (upper and lower). Let's explain what we mean by that.

Upper and Lower Bounds

In the previous tutorial (2.1) we discussed "rounding" which is a method used to express a number in a simpler way by appointing it an approximate value that is easier to remember.

For example, when we have a set of 237 items, we might say, "there are about 240 items" or "there are about 200 items" in the set. In such cases, we express the amount of items through a number that is easier to remember, despite knowing that this number does not represent the exact amount of items in the set.

This method (rounding) is useful for estimating but it does have a drawback. We don't know whether there are less or more items in the set than the approximate number used for rounding. For example, if we say "you have about 6 hours from now to complete a given task", this deadline may range between 5 hours and 30 minutes and 6 hours and 30 minutes from now. Hence, rounding is useful as method of expressing numbers but it also leaves place for misinterpretation.

To address this issue, is it worth confirming the range of values that the rounded number may extend to in addition to the rounded value of the known number. For example, if one says "a number rounded to the nearest ten is 70", we must immediately think about the range of possible values that have this rounding. Obviously, this is an example of rounding to the nearest ten, so the possible values that when rounded to the nearest ten become 70 are: 65, 66, 67, 68, 69, 70, 71, 72, 73 and 74. The smallest from these number (65) is known as the lower bound, while the greatest of them (74) is known as the upper bound.

By definition, the lower bound is the smallest actual value that gives a certain rounded number, while the upper bound is the largest actual value that gives the same rounded number.

Example 1

Determine the lower and upper bound of numbers below when

  1. The rounded value of a given number to the nearest ten is 350
  2. The rounded value of a given number to the nearest hundred is 600.

Solution 1

  1. If an unknown number gives 350 after having been rounded to the nearest ten, its possible values range from 345 (the lower bound) to 354 (the upper bound).
  2. If an unknown number gives 600 after having been rounded to the nearest hundred, its possible values range from 550 (the lower bound) to 649 (the upper bound).

Example 2

All numbers below share a common property except one of them. Identify this number and the common property of the others.

765; 850; 1249; 514; 823

Solution 2

765 is a lower bound for a set of numbers that give 770 when rounded to the nearest ten.

850 is a lower bound for a set of numbers that give 900 when rounded to the nearest hundred.

1249 is an upper bound of a set of numbers that give 1200 when rounded to the nearest hundred.

514 is an upper bound of a set of numbers that give510 when rounded to the nearest ten.

823 is neither a lower nor an upper bound for any set of numbers. Therefore, it is different from the rest of numbers.

We can use approximations made during rounding process to estimate quickly the rough value of an expression. Let's see an example:

Example 3

Estimate quickly the value of expression of

18.2 × 251.4/2.517

by rounding each number to two significant figures.

Solution 3

We use rounding to write each number in two significant figures. Thus, we must write 18.2 as 18, 251.4 as 250 and 2.517 as 2.5. Therefore, we have

18 × 250/2.5 = 18 × 100 = 1800

The exact value is not very different from the one obtained above. Indeed, we have

18.2 × 251.4/2.547 = 1817.83

The concepts of upper and lower bound are applied in practice to know the minimum and maximum value of an item. Let's explain this point through an example.

Example 4

A customer is interested in painting a room and so, he calls a painting company. When asked about the dimensions of the room, he responds: "The room is about 6 m long, about 5 m wide and about 4 m high." Giving that 5 m2 of surface require 1 kg of paint, what is the minimum and maximum amount of paint to be used in this process? Use decimetre as the smallest unit of measurements. (The floor is not painted).

Solution 4

It is clear that all dimensions are rounded to the nearest metre. Thus, the values of lower bounds are Lmin = 5.5 m, W = 4.5 m and H = 3.5 m, where L, W and H stand for length, width and height respectively.

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

On the other hand, the maximum values for these dimensions are L = 6.4 m, W = 5.4 m and H = 4.4 m.

Since the area of each face is the product of the two corresponding dimensions, we obtain for the surface area A to be painted:

A = 2 × L × H + 2 × W × H + L × W

The minimum area to be painted therefore is

Amin = 2 × (5.5 m × 3.5 m) + 2 × (4.5 m × 3.5 m) + (5.5 m × 4.5 m)
= 38.5 m2 + 31.5 m2 + 24.75 m2
= 94.75 m2

Hence, the minimum amount of paint Pmin needed is

Pmin = 94.75 m2/5 m2/kg = 18.95 kg paint

As for the maximum amount of paint needed for the room, we have

Amin = 2 × (6.4 m × 4.4 m) + 2 × (5.4 m × 4.4 m) + (6.4 m × 5.4 m)
= 56.32 m2 + 47.52 m2 + 34.56 m2
= 138.4 m2

As you see from the above examples, the lower and upper bound of a value are expressed in a value that is of one position higher precision that the rounded value. For example, if the rounded value is expressed in tens, the upper and lower bounds are expressed in units, when the rounded value is expressed in units the bounds are expressed in tenths and so on. Let's illustrate this point though an example.

Example 5

Calculate the minimum and maximum speeds of a car travelling 250 km in 4 hours if both distance d and time t represent rounded values. Express the results in three significant figures.

Solution 5

Considering the concepts of lower and upper bounds, we have:

dmin = 245 km
dmax = 254 km
tmin = 3.5 h
tmax = 4.4 h

Giving that speed = distance / time, it is clear that the minimum speed is obtained when dividing the minimum distance and maximum time, while the maximum speed is obtained when dividing the maximum distance and minimum time. Therefore, we have

vmin = dmin/tmax = 245 km/4.4 h = 55.7 km/h

and

vmax = dmax/tmin = 254 km/3.5 h = 72.6 km/h

Note the big difference between these two values dictated by the fact that the original values were not so precise. In this way, a wide range of possible values is produced.

Intervals and Segments

An Interval is a set of numbers in which all values are included except the bounds. It is expressed through the curved brackets, (). For example, if we see the following written somewhere (3, 8), we read "the interval that includes all values between 3 and 8 without these two bounds."

A Segment on the other hand, represents a set of numbers that besides the in-between values includes the two bounds as well. A segment is expressed through the square brackets [ ]. Thus, if we see written [4, 11], we read "the segment that includes all values between 4 and 11 including the bounds."

Intervals are shown in the number line by white (empty) dots, while segments by black (filled) dots. Look at the figure below.

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

We say the ends of an interval are open, while those of a segment are closed.

On the other hand, when the lower bound (the left end) is open and the upper bound (the right end) if closed, we have a half-interval. Likewise, when the lower bond (the left end) is closed and the upper bound (the right end) is open, we have a half-segment. As you see the first (left) bound gives the name to the structure. Look at the figure below.

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

Example 6

Express the number sets (-2, 7], [1, 4], (0, 5) and [1, 8) in the number line.

Solution 6

(-2, 7] represents a half-interval. It is open at the left and closed at the right end. It is shown in the number line as

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

[1, 4] is a segment. It is closed at both sides and is shown in the number line as

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

(0, 5) is an interval. It is open at both ends and is represented in the number line as follows

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

[1, 8) is a half-segment. It is closed at the left end and open at the right one.

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

When one end of a number set extends to infinity, we represent that part using the symbol of interval, as it is impossible to find the exact value of infinity. For example, the set that extends from 4 to infinity (including 4), is a half segment that is symbolically written as [4, +∞), where ∞ is the symbol of infinity. When shown in the number line, the part that goes towards infinity is open, as shown in the figure.

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

The same thing occurs when the interval points toward negative infinity. In this case, the arrow points toward left. For example, if we see written (-∞, 6], we read "the set of numbers extending from minus infinity to 6, including the number 6" or simply "the half-interval from minus infinity to 6".

Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

Example 7

Show the following number sets in the number line.

  1. [-3, +∞)
  2. (-∞, 2)
  3. (-∞, +∞)

Solution 7

  1. We have a half-segment that starts from -3 and extends on the right towards positive infinity. Hence, when this set is shown in the number line, we get Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  2. We have an interval that starts from negative infinity and extends to the right up to the number 2, without including this number. Hence, when this set is shown on the number line, we getMath Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  3. This set extends to infinity on both sides, so it is indicated by a double-sided arrow in the number line. Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example

Rounding through Intervals and Segments Explained

Intervals and segments can be used to explain rounding. More specifically, we use the concept of half-segment to indicate the possible values of a number which, when rounded, provides the same value. In this case, the number included in the lower bound is included in the set while, after the upper bound, there is always an "uncovered" part which is dictated by the precision of numbers involved.

For example, if a number is 30 when rounded to the nearest ten, we have taken the lower bound as 25 and the higher bound as 34. But when the rounded number is 40, this includes values from 35 to 44. It is clear that the part from 34 to 35 remains uncovered as long as we don't increase the precision of numbers to include non-whole values in the set.

To avoid these issues, we introduce the concept of a "half-segment" to represent the possible range of a rounding. We say "all numbers included in the half-segment [25, 34) are 30 when rounded to the nearest ten, this includes non-whole numbers". In this way, we no longer need to specify the set of numbers involved; all number sets have the same structure of representation when rounding to the nearest ten. The same can be said for other types of rounding (to the nearest hundred, thousand, unit, tenths, thousandths, etc.).

Example 8

Show, using symbols and number line, the set of numbers that are:

  1. 70 when rounded to the nearest ten.
  2. 400 when rounded to the nearest hundred.
  3. 9000 when rounded to the nearest thousand.
  4. 14 when rounded to the nearest unit.
  5. 5.6 when rounded to the nearest tenth.

Solution 8

  1. The numbers that are 70 when rounded to the nearest ten extend from 65 (including it) to 75 (without including it). Hence, when represented by symbols, this set is written as [65, 75). When shown in a number line, this set gives Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  2. The numbers that are 400 when rounded to the nearest hundred extend from 350 (including it) to 450 (without including it). Hence, when represented by symbols, this set is written as [350, 450). When shown in a number line, this set gives Math Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  3. The numbers that are 9000 when rounded to the nearest thousand extend from 8500 (including it) to 9500 (without including it). Hence, when represented by symbols, this set is written as [8500, 9500). When shown in a number line, this set givesMath Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  4. The numbers that are 14 when rounded to the nearest unit extend from 13.5 (including it) to 14.5 (without including it). Hence, when represented by symbols, this set is written as [13.5, 14.5). When shown in a number line, this set givesMath Tutorials: Upper and Lower Bounds. Intervals and Segments Example
  5. The numbers that are 5.6 when rounded to the nearest tenth extend from 5.55 (including it) to 5.65 (without including it). Hence, when represented by symbols, this set is written as [5.55, 5.65). When shown in a number line, this set givesMath Tutorials: Upper and Lower Bounds. Intervals and Segments Example Remark! 5.60 and 5.6 represent the same number; the extra zero simply provides a higher precision in measurement.

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