Mathematics
Rounding
Rounding is the process of approximating a number to a certain place value, such as a whole number or a specific decimal place. When rounding, numbers are adjusted to make them easier to work with or understand. This is commonly used in everyday calculations and in situations where precise values are not necessary.
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No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Rounding Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing US$23.4476 with US$23.45, or the fraction 312/937 with 1/3, or the expression with 1.41. Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may be done also to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as about 123,500. On the other hand, Rounding introduces some round-off error in the result. Rounding is almost unavoidable in many computations — especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating point representation with a fixed number of significant digits. In a sequence of calculations, these Rounding errors gene-rally accumulate, and in certain ill-conditioned cases they may make the result mea-ningless. Accurate Rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as the table-maker's dilemma. Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals. Types of Rounding Typical Rounding problems are • approximating an irrational number by a fraction, e.g. π by 22/7; • approximating a fraction with periodic decimal expansion by a finite decimal fraction, e.g. 5/3 by 1.6667; • replacing a rational number by a fraction with smaller numerator and deno-minator, e.g. 3122/9417 by 1/3;
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Rounding Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing US$23.4476 with US$23.45, or the fraction 312/937 with 1/3, or the expression with 1.41. Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may be done also to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as about 123,500. On the other hand, Rounding introduces some round-off error in the result. Rounding is almost unavoidable in many computations — especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating point representation with a fixed number of significant digits. In a sequence of calculations, these Rounding errors generally accumulate, and in certain ill-conditioned cases they may make the result meaningless. Accurate Rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as the table-maker's dilemma. Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals. Types of Rounding Typical Rounding problems are • approximating an irrational number by a fraction, e.g. π by 22/7; • approximating a fraction with periodic decimal expansion by a finite decimal fraction, e.g. 5/3 by 1.6667; • replacing a rational number by a fraction with smaller numerator and denominator, e.g. 3122/9417 by 1/3;
- Khalid Khan, Tony Lee Graham(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
1 Review of Basic Concepts1.1 Degrees of Accuracy1.1.1 Rounding Numbers (Common Method)In the real world, when dealing with numbers a degree of accuracy is needed. For example, if a piece of wood of a certain length was required, asking for a length of 123.732461 centimeters would not be sensible as such accurate measurements are not possible. What is more usual is some form of Rounding. The method of Rounding is commonly used in mathematical applications in science and engineering. It is the one generally taught in mathematics classes in high school. The method is also known as round-half-up . It works as follows:- Decide which is the last digit to keep.
- Increase it by 1 if the next digit is 5 or more (this is called Rounding up).
- Leave it the same if the next digit is 4 or less (this is called Rounding down).
Example 1.1- 3.044 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
- 3.045 rounded to hundredths is 3.05 (because the next digit, 5, is 5 or more).
- 3.0447 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
For negative numbers, one rounds the absolute value and reapplies the sign afterward.Example 1.2- −2.1349 rounded to hundredths is −2.13.
- −2.1350 rounded to hundredths is −2.14.
1.1.2 Round-to-Even MethodThe round-to-even method method, also known as unbiased Rounding or Gaussian Rounding , exactly replicates the common method of Rounding except when the digit(s) following the Rounding digit starts with a 5 and has no nonzero digits after it
eBook - PDF- Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
1 NASA images/Shutterstock.com Chapter 1 Arithmetic Operations Used in Statistics CHAPTER CONTENTS Section 1.1 Rounding Numbers Section 1.2 Types of Numbers and the Number Line Section 1.3 Fractions, Decimals, and Percentages Section 1.4 Operations with Fractions Section 1.5 Absolute, Relative, and Percent Error Section 1.6 Scientific Notation and E-notation Section 1.7 Read and Use Mathematical Tables 2 CHAPTER 1 • Arithmetic Operations Used in Statistics SECTION 1.1 Rounding Numbers Most of us on a daily basis deal with Rounding numbers in a variety of ways. If some-one asks you how much longer a certain task will take, you might reply, “About 15 minutes.” If it is a major project, your reply might be “about 4 hours” or even “about 2 weeks.” When giving these different approximations for the time required to complete a task, we are providing an appropriate frame of reference (“minutes,” “hours,” or “weeks”) to the project. We are supplying an implied level of accuracy that both individuals should find appropriate. Likewise, in mathematics and statistics, we often will be asked to round a number. In these situations, we write a numerical value with an agreed accuracy. This prac-tice allows us to avoid writing long and drawn-out decimal numbers. The appropri-ate application of Rounding, however, depends on an agreement on how we go about Rounding. OBJECTIVE 1 Round Decimal Numbers Many times, we need to round numbers that are relatively small. By “small,” we mean numbers whose values are between 0 and 1. Numbers such as 0.2548 or 0.000361 begin with a zero, include a decimal point, and then include digits to the right of the decimal point. We will refer to these numbers as decimal numbers . Decimal numbers are important in the study of statistics and arise quite frequently. Now let’s determine how we round a decimal number to an agreed decimal place.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
(22% of the fourth graders provided a correct answer, 26% answered 90, and 48% gave another number.) This performance shows that interpreting a rounded result needs to be carefully addressed. It would be valuable to provide opportunities for children to reflect on how to interpret the result of a rounded number. For example, what are the smallest and largest integers that could be rounded to get 1300. Figure 8-21 provides visual reminder of the range that would include any number from 1250 to 1350. A n important aspect of developing number sense is rec- ognizing that some numbers are approximate (such as our national debt) and some are exact (such as the number of people killed in a fatal airplane crash). Approximate values are associated with estimation, often involving Rounding, and are encountered regularly in our daily lives. Rounding integrates understanding of approximate val- ues with place value and naming numbers. Numbers are usu- ally rounded to make them easier to use or because exact values are unknown. For example, Rounding off and round- ing up are two very direct and easy strategies for making numbers easier to handle. Some examples are shown in Figure 8-20. The round off procedure is a reminder of our earlier dis- cussion of front-end digits. In every case, the round off pro- duces the front-end digit, whereas the round-up produces a value that is the next digit higher and generally has the same place value. (The exception being when 92 rounds up to 100.) Round off and round up are quick and easy to do. It places bounds on the value, but the range of values of the original number can be great. For example, any value between 7000 and 7999 would round off to 7000 and round up to 8000. How numbers are rounded depends on how they are used. For example, in buying something for $8.25, you would round up to $9 to make sure you had enough money. Attendance of 54,321 at a major league baseball game pro- vides a different context.
eBook - PDFMathematical Literacy NQF3 SB
TVET FIRST
- K van Niekerk O Roberts(Author)
- 2017(Publication Date)
- Macmillan(Publisher)
What is the mistake here? b) Now calculate what the price difference would be if 2 000 dresses were sold with the same Rounding error. 2. To calculate the actual distance on a map of scale 1 : 1 000 000, you need to multiply the measured distance by 1 000 000 and then convert to kilometres. a) Sara measures a distance of 54 mm on the map and rounds off to 50 mm. What is the Rounding off error in the actual distance? b) What is the Rounding off error in the actual distance if 3,8 cm is rounded off to 4 cm? 3. Tebogo wants to calculate the area of a circle which has a radius of 4,43 cm. She has to give the answer rounded to one decimal place. She uses the formula: Area = π × r 2 and uses 3,14 for the value of π. a) She rounds off the radius to one decimal place before doing the calculation. Calculate her answer. b) What answer would she get if she rounded off to one decimal place after calculating the area? 13 Module 1 Work with numbers correctly Unit 1.4: Applying addition and multiplication facts Calculators and computers make our lives much easier, but we still need to be able to calculate quickly and on the spot. For example, if you need to check the size of a space required to unload and store stock, or to check the quantities of materials required to complete a work task, you should be able to do so without a calculator. Estimating Estimation is the method of finding approximate answers to a problem in a short time. We can use estimation when we are in a situation when we cannot do the full calculation, or to check the answers to a calculation. Some uses of estimation include: l quickly checking to see if you have enough money to buy what you want l getting a rough idea of the correct answer to a problem l checking that you have entered information into a calculator correctly. Rounding off numbers is a good way to estimate. Example 1.6 Estimate the cost of 15 books at R95,95. Solution Round off the cost to R100 to get 15 × R100 = R1 500.
eBook - PDFComputing Methods
Adiwes International Series in the Engineering Sciences
- I. S. Berezin, N. P. Zhidkov(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
5. He must watch the accuracy of the intermediate results in the process of calculation so as to ensure the necessary accuracy of the final result on the one hand, and to simplify the computa-tions as far as possible on the other. The last two points are very important. Krylov pointed out that he had had to examine pro-jects in which 90 per cent of the work had been wasted on the com-putation of unnecessary or wrong figures - all of this because the rules for working with approximate quantities were not known! (3) Rules for Rounding numbers Let us agree on certain terminology before turning to the study of these rules. It is assumed that the numbers with which we shall have to deal can always be written in terms of a finite number of digits in one or another number system. Thus, if we base the number system on the natural number ß and exclude numbers with more than m digits, we can write them uniquely in the form: ± ( α ^ + α ^ -1 * . . . +α 7η ^-^~ 1 ), where α· are positive whole numbers, O^ai^ß. It is of interest to note that the method used here is the one of which we spoke in the Introduction. Instead of the whole set of actual numbers in a certain interval, use is made of a finite discrete sub-set. It is some- 4 COMPUTING METHODS times necessary and desirable in the process of calculation to go beyond the limits of this sub-set, but the number of digits fre-quently remains limited and we still have to deal with a finite set of numbers. It may be that the numbers of digits in the result is infinite or very large so that they cannot be contained in the machine, or else they are too cumbersome for calculations on paper. Then we must substitute a certain number from our basic sub-set for the result. We naturally take the nearest number in this sub-set. In practice this entails the following. Suppose we obtain the following number as the result of a calculation: ± ( α ^ + α ^ -ι * . . . +a m ß«-™+ 1 + a m+l ß n -' n + . . .).
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