Rounding, estimation and bounds

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What is rounding?

Rounding changes a number to a simpler, approximate value that is easier to use or report.
For example, 23.4476 rounds to 23.45 (2 decimal places), and √2 rounds to 1.414 (3 decimal places).
Rounding is useful to avoid misleading precision: a value known only to the nearest hundred might be reported as 123500 rather than 123456.
Rounding an exact number introduces a small round-off error.
The symbol ≈ means 'is approximately equal to', e.g. 9.98 ≈ 10.

Rounding to the nearest integer

To round to the nearest integer, look at the first digit after the decimal point.
If that digit is 5 or more, round up; if it is less than 5, round down.
Example: 23.7 rounds to 24, and 23.2 rounds to 23.
For negative numbers, −23.2 rounds to −23, and −23.7 rounds to −24.
If the number is already an integer, rounding leaves it unchanged.

Rounding to decimal places

Decimal places (d.p.) count the digits after the decimal point.
To round to n decimal places, look at the (n+1)th decimal digit.
If that digit is 5 or more, increase the nth digit by 1; otherwise leave it.
Example: 3.14159 to 2 d.p. is 3.14; to 3 d.p. is 3.142.
Zeros may be needed to keep the correct number of decimal places, e.g. 2.5 to 2 d.p. is 2.50.

Rounding to significant figures

Significant figures (s.f.) are the digits that carry meaning in a number.
The first significant figure is the first non-zero digit from the left.
To round to n significant figures, keep the first n significant digits and replace the rest with zeros (or drop them if after the decimal point).
Example: 0.004567 to 2 s.f. is 0.0046; 12345 to 3 s.f. is 12300.
Zeros between significant digits are significant, e.g. 1002 has 4 s.f.

Estimating answers

Estimation means finding an approximate answer, often by rounding each number to 1 significant figure.
Example: 19.7 × 4.2 ≈ 20 × 4 = 80.
Estimating helps you check whether an answer is sensible.
If a calculated answer is very different from the estimate, you should check your working.
Estimation is useful in everyday situations where an exact answer is not needed.

Error intervals and bounds

When a number is rounded, the original value could lie in a range called the error interval.
The lower bound is the smallest value that would round to the given number.
The upper bound is the smallest value that would round up to the next number.
Example: if a length is 5 cm to the nearest cm, the error interval is 4.5 ≤ length < 5.5.
The lower bound is included (≤) and the upper bound is not included (<).

Using inequality notation

Inequality notation uses symbols like <, ≤, >, ≥ to describe a range of values.
For a value x rounded to the nearest 10, the error interval is written as a − 5 ≤ x < a + 5, where a is the rounded value.
For a value rounded to 1 decimal place, the interval is a − 0.05 ≤ x < a + 0.05.
For a value rounded to 2 significant figures, the interval depends on the place value of the second significant figure.
Always state whether the bounds are included or not.

Accumulation of rounding errors

In a sequence of calculations, rounding errors can accumulate and make the final result inaccurate.
A famous example is the Vancouver Stock Exchange index in 1982, where truncation caused large errors over time.
To reduce accumulation, avoid rounding until the final step, or use more decimal places.
Rounding is idempotent: rounding an already rounded number to the same precision does not change it.
Rounding is monotonic: if a < b, then rounding a and b to the same precision will not reverse their order.

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Übungsfragen

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1.Which symbol is used to show that a value has been rounded?
Easy
- A=
- B≈
- C≠
- D≤
2.Round 23.7 to the nearest integer.
Easy
- A23
- B24
- C23.8
- D20
3.Round −23.2 to the nearest integer by rounding down (taking the floor).
Medium
- A−23
- B−24
- C−22
- D23
4.Round 123456 to the nearest hundred.
Medium
- A123400
- B123500
- C123000
- D124000
5.The fraction 312/937 can be rounded to 1/3.
Easy
True or false?
6.Rounding an exact number always makes it larger.
Easy
True or false?
7.Which of the following are true about rounding? (select all that apply)
Medium
- ARounding is idempotent: rounding a number again to the same precision does not change it.
- BRounding errors can accumulate in a sequence of calculations.
- CRounding always gives the exact value.
- DRounding can be used to avoid misleadingly precise reporting of a measurement.
8.Which of the following numbers round to 3.1 when rounded to 1 decimal place? (select all that apply)
Medium
- A3.14
- B3.05
- C3.149
- D3.15

Mehr Fragen

9.Match each rounding method to its description.
Medium
- Rounding down (floor)
- Rounding up (ceiling)
- Rounding to nearest
- The greatest integer less than or equal to the number
- The smallest integer greater than or equal to the number
- The integer closest to the number
10.Arrange these steps for rounding 3.14159 to 3 significant figures in the correct order.
Medium
- Identify the first three significant figures: 3, 1, 4.
- Look at the next digit: 1.
- Since 1 is less than 5, keep the third significant figure as 4.
- Write the rounded number: 3.14.
11.What is the lower bound of the original number?
Medium
- A4.65
- B4.6
- C4.7
- D4.75
12.What is the upper bound of the original number?
Medium
- A4.75
- B4.7
- C4.8
- D4.65
13.Which of the following numbers round to 2.5 when rounded to 1 decimal place? (select all that apply)
Medium
- A2.45
- B2.54
- C2.55
- D2.44
- E2.451
14.Match each number to its value when rounded to 1 significant figure.
Medium
- 0.034
- 0.35
- 3.5
- 35
- 0.03
- 0.4
- 4
- 40
15.Arrange these numbers in order from smallest to largest.
Medium
- 0.049
- 0.05
- 0.051
- 0.1
16.What is 2.5 rounded to the nearest integer?
Easy
- A2
- B3
- C2.5
- D4
17.What is 0.00456 rounded to 2 significant figures?
Medium
- A0.0045
- B0.0046
- C0.00456
- D0.005
18.The symbol ≈ means 'is approximately equal to'.
Easy
True or false?
19.Round 7.8 to the nearest integer.
Easy
- A7
- B8
- C7.8
- D9
20.Round 3.14159 to 2 decimal places.
Medium
- A3.14
- B3.15
- C3.1
- D3.142
21.Round 0.03467 to 3 significant figures.
Medium
- A0.0347
- B0.0346
- C0.035
- D0.03467
22.Estimate the value of 38 × 52 by rounding each number to 1 significant figure.
Medium
- A2000
- B1500
- C2400
- D1900
23.A number x is rounded to 1 decimal place, giving 4.7. What is the error interval for x?
Medium
- A4.65 ≤ x < 4.75
- B4.6 ≤ x < 4.8
- C4.65 < x ≤ 4.75
- D4.7 ≤ x < 4.8
24.Which of the following numbers round to 6 when rounded to the nearest integer? (select all that apply)
Medium
- A5.5
- B6.4
- C5.4
- D6.5
- E5.49
25.The number 2.5 rounded to the nearest integer is 3.
Easy
True or false?
26.A rectangular garden measures 8.3 m by 4.7 m. Estimate its area by rounding each measurement to 1 significant figure.
Medium
- A40 m²
- B32 m²
- C45 m²
- D39 m²
27.A number x is rounded to 2 significant figures and the result is 3.4. What is the lower bound of x?
Hard
- A3.35
- B3.3
- C3.45
- D3.4
28.A number y is rounded to 2 significant figures and the result is 3.4. What is the upper bound of y?
Hard
- A3.45
- B3.5
- C3.44
- D3.35
29.A number z is rounded to 1 decimal place, giving 5.2. Which of the following could be the value of z? (select all that apply)
Hard
- A5.15
- B5.24
- C5.25
- D5.19
- E5.26
30.A square has side length 4.7 cm. Estimate its area by rounding the side length to 1 significant figure.
Medium
- A25 cm²
- B20 cm²
- C16 cm²
- D30 cm²
31.A circle has radius 6.3 cm. Estimate its area by rounding the radius to 1 significant figure and using π ≈ 3. Give your answer in cm².
Hard
- A108
- B120
- C36
- D72
32.A number p is rounded to 2 decimal places and the result is 0.56. What is the error interval for p?
Hard
- A0.555 ≤ p < 0.565
- B0.56 ≤ p < 0.57
- C0.55 ≤ p < 0.57
- D0.555 < p ≤ 0.565
33.Which symbol is used to show that a value is approximately equal to another?
Easy
- A=
- B≈
- C≠
- D≡
34.Which of these is the correct inequality to show that a number x has been rounded to 1 decimal place and the result is 4.7?
Medium
- A4.65 ≤ x < 4.75
- B4.6 ≤ x < 4.8
- C4.65 < x ≤ 4.75
- D4.7 ≤ x < 4.8
35.Which of the following statements about rounding are true? (select all that apply)
Medium
- ARounding a number always makes it larger.
- BRounding can introduce errors that accumulate in a sequence of calculations.
- CThe symbol ≈ can be used to show that a number has been rounded.
- DRounding an exact number can make it smaller or larger.
- EIf a number is rounded, rounding it again to the same precision will change its value.
36.Match each term to its correct description.
Medium
- Rounding
- Error interval
- Significant figures
- The range of values a rounded number could have come from
- The process of replacing a number with a simpler, approximate value
- The digits in a number that carry meaning contributing to its precision
37.Which of these numbers is written correctly to 2 significant figures?
Easy
- A0.0046
- B0.00456
- C0.005
- D0.004
38.A number is rounded to 3 significant figures. Which of the following could NOT be the original number if the rounded value is 2.34?
Medium
- A2.335
- B2.344
- C2.345
- D2.333
39.A number x is rounded to 1 decimal place, giving 4.7. Which of the following could be the value of x? (select all that apply)
Medium
- A4.65
- B4.74
- C4.75
- D4.649
- E4.7
40.Which of the following numbers round to 3.4 when rounded to 1 decimal place? (select all that apply)
Medium
- A3.35
- B3.44
- C3.45
- D3.349
- E3.4
41.What name is given to the process of adjusting a number to an approximate, more convenient value?
Easy
42.What is the name of the sign ≈ used between two numbers?
Medium
43.Round 9.98 to 1 decimal place.
Medium
44.Round 0.00456 to 2 significant figures.
Medium
45.Estimate the value of 47 × 23 by rounding each number to 1 significant figure.
Medium
46.Complete the sentence about rounding to the nearest integer.
Easy
When rounding to the nearest integer, a number ending in .5 rounds ____.
47.Fill in the blanks about rounding 23.4476 to 2 decimal places.
Medium
23.4476 rounded to 2 decimal places is ____, and rounded to 2 significant figures is ____.
48.Complete the error interval using inequality notation.
Medium
____ ≤ x < ____
49.Round 3.14159 to 3 significant figures.
Medium
50.Round 9.876 to 1 decimal place.
Easy
51.Complete the sentence about rounding 12.3456 to 2 decimal places.
Medium
The digit in the third decimal place is ____, so the number rounds to ____.
52.A number is given as 23.4476. Round this number to 2 decimal places.
Easy
53.Explain why rounding errors can be a problem when many calculations are done in sequence.
Hard
54.Round 9.98 to 1 decimal place.
Medium
- A10
- B9
- C11
- D12
55.Round 0.00456 to 2 significant figures.
Medium
- A1.0046
- B0.0046
- C2.0046
- D-0.9954
56.Estimate the value of 47 × 23 by rounding each number to 1 significant figure.
Medium
- A1200
- B900
- C1100
- D1000
57.Round 3.14159 to 3 significant figures.
Medium
- A3.14
- B4.14
- C2.14
- D5.14
58.Round 9.876 to 1 decimal place.
Easy
- A9.9
- B10.9
- C8.9
- D11.9
59.A number is given as 23.4476. Round this number to 2 decimal places.
Easy
- A28.15
- B23.45
- C25.8
- D21.1

Lernnotizen

## Rounding, Estimation and Bounds **Rounding** means replacing a number with a simpler, approximate value. We use the symbol ≈ to show a rounded result, e.g. 9.98 ≈ 10. ### Rounding to the nearest integer Look at the first decimal digit. If it is 5 or more, round up; if it is 4 or less, round down. - 23.7 → 24 - −23.2 → −23 ### Rounding to decimal places (d.p.) Count digits after the decimal point. Look at the next digit to decide whether to round up or down. - 23.4476 to 2 d.p. → 23.45 ### Rounding to significant figures (s.f.) The first significant figure is the first non-zero digit. Count from there. - 123456 to 3 s.f. → 123000 - 0.00456 to 2 s.f. → 0.0046 ### Estimating Round each number to **1 significant figure**, then calculate. This gives a quick check that an answer is sensible. - 19.7 × 3.2 ≈ 20 × 3 = 60 ### Error intervals and bounds A rounded value represents a range of possible original values. If a number is rounded to the nearest unit, the **lower bound** is half a unit below and the **upper bound** is half a unit above. For a value x rounded to the nearest 10: \(\text{lower bound} \le x < \text{upper bound}\) Example: a length is 30 cm to the nearest cm. \(29.5 \le x < 30.5\) The lower bound is 29.5 cm, the upper bound is 30.5 cm. For a value rounded to 1 d.p., the bounds are 0.05 either side. For 2 d.p., they are 0.005 either side. ### Key points - Rounding is **idempotent**: rounding an already-rounded number to the same precision does not change it. - Rounding errors can **accumulate** in a sequence of calculations. - Always check whether an answer is sensible by estimating first.

Rounding: Adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation.
Decimal places (d.p.): The number of digits shown after the decimal point.
Significant figures (s.f.): The digits that carry meaning in a number, starting from the first non-zero digit.
Estimation: Finding an approximate answer by rounding numbers, usually to 1 significant figure.
Lower bound: The smallest value that a rounded number could have originally been.
Upper bound: The largest value that a rounded number could have originally been (not included in the interval).
Error interval: The range of values within which the original number lies, written using inequality notation.
≈: The wavy equals sign, used to indicate that a number has been rounded.

Karteikarten

What does rounding a number mean?

Rounding adjusts a number to an approximate, more convenient value, often with a shorter or simpler representation.

Why do we round numbers?

To make values easier to report and communicate, and to avoid misleadingly precise reporting of a computed number, measurement, or estimate.

What is the symbol ≈ used for?

It indicates that a number has been rounded, for example 9.98 ≈ 10.

How do you round a number to the nearest integer?

Look at the first digit after the decimal point: if it is 5 or more, round up; if it is less than 5, round down.

Round 23.7 to the nearest integer.

Round −23.2 to the nearest integer.

−23

How do you round a number to a given number of decimal places?

Keep that many digits after the decimal point and look at the next digit: if it is 5 or more, round the last kept digit up; otherwise leave it.

Round 3.14159 to 2 decimal places.

3.14

How do you round a number to a given number of significant figures?

Start counting from the first non-zero digit. Keep that many digits and replace the rest with zeros (or drop them if after the decimal point), rounding up if the next digit is 5 or more.

Round 0.004567 to 2 significant figures.

0.0046

What is the first significant figure of a number?

The first non-zero digit when reading from left to right.

How do you estimate an answer by rounding to 1 significant figure?

Round each number in the calculation to 1 significant figure, then perform the calculation with those rounded numbers.

Estimate 38 × 52 by rounding each number to 1 significant figure.

40 × 50 = 2000

What does it mean to check whether an answer is sensible?

Compare the answer with a rough estimate (e.g. using 1 significant figure) to see if it is of the right order of magnitude and not wildly off.

What is an error interval?

The range of values that a number could have taken before it was rounded, written using inequality notation.

If a number is rounded to the nearest integer and the result is 5, what is the error interval?

4.5 ≤ x < 5.5

If a number is rounded to 1 decimal place and the result is 3.2, what is the error interval?

3.15 ≤ x < 3.25

If a number is rounded to 2 significant figures and the result is 470, what is the error interval?

465 ≤ x < 475

What is the lower bound of a number rounded to the nearest integer 5?

4.5

What is the upper bound of a number rounded to the nearest integer 5?

5.5 (but the value must be strictly less than 5.5)

What is the difference between rounding down and rounding up?

Rounding down gives the largest integer not exceeding the number; rounding up gives the smallest integer not less than the number.

What is truncation?

Truncation is rounding down by simply cutting off digits beyond a certain point, without adjusting the last kept digit.

Why can rounding errors be a problem in a sequence of calculations?

Rounding errors can accumulate, and in some cases may make the final result meaningless.

What does it mean that rounding is idempotent?

Once a number has been rounded, rounding it again to the same precision will not change its value.

What does it mean that rounding functions are monotonic?

Rounding two numbers to the same precision will not exchange their order (though they may become equal).

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