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How to Round Numbers

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How to Round Numbers and Understand Significant FiguresIntroductionRounding numbers is a important mathematical skill that is often used in everyday life. For example, when estimating how much money you'll need to spend on groceries for the week, you'll likely round the total cost up or down to the nearest dollar. Significant figures are another important concept in mathematicsthat is often used in real-world situations. For instance, when measuring the length of a room, the number of significant figures in your measurement will determine the accuracy of your estimate.Rounding Numbers.When to Round NumbersIn mathematics, rounding is a technique used when approximating a number or value. The general idea is to select the closest number that is still less than or equal to the original number. There are many different situations in which rounding numbers may be necessary. For example, you may need to round numbers when:-Estimating: When you are making an estimate, you will want to round up or down to the nearest value that you are confident in. This will give you the most accurate estimate possible without going into too much detail.-Calculating: There are often times when calculating numbers that it is not necessary to get an exact answer. In these cases, rounding can simplify the calculation and make it easier to determine the answer.-Reporting results: When reporting results from experiments or data collected, it is often helpful to round numbers so that they are easier for others to understand and interpret.Significant Figures.Significant figures are the digits in a number that are used to express its precision. The concept of significant figures is important in science and engineering because it allows us to communicate the uncertainty of measurements.There are two types of significant figures:1) those that express the precision of measurements (measurement significant figures), and2) those used in calculations (calculation significant figures).The number of significant figures in a measurement is determined by the smallestunit of measure being used. For example, if you are measuring something using a ruler that is marked in centimeters, then the smallest unit of measure is 1 centimeter (cm). This means that your measurement can only be expressed to the nearest centimeter. Therefore, the number of significant figures in your measurement would be limited to 2 (the ones and tens place).The number of significant figures in a calculation is determined by the number with the least amount of digits. For example, if you are adding up two numbers: 3456 + 789 = 4245. In this case, the number with the least amount of digits is 789, which has 3 digits. Therefore, the answer can only be expressed to 3 significant figures, and would be rounded to 4250.Why are Significant Figures ImportantAs we mentioned before, significant figures are important because they allow us to communicate the uncertainty of measurements. When we make a measurement, there is always some degree of error involved. The more precise our measurement is, the smaller our margin of error will be. However, even with very precise measurements there will still be some uncertainty due to factors such as human error or instrument error.For instance, let's say you measure the length of an object using a ruler that is marked in centimeters (cm). You might get a reading of 10 cm for your first try. But if you measure it again, you might get a slightly different reading such as 9.9 cm or 10.1 cm. These readings would all be considered accurate because they are within the same range (10 cm +/- 0.1 cm). But if your second try yielded a reading of 11 cm, then this would be considered inaccurate becauseit is outside of the range specified by your first measurement (+/- 0 .1cm).So how do we know how precise our measurements are? This brings us back to why significant figures are important… by understanding sig figs we can estimate just how precise our measurements really are!For example let's say you measure an object to be 10 cm long with a ruler that has markings for every centimeter (smallest unit = 1cm). In this case you would report your measurement as 10 cm +/- 1cm because that is the smallest unit on your ruler and thus limits your precision to 1cm.* So even though you got an exact reading of 10 cm, you know that your real length could actually be anywhere between 9-11 cm.*This assumes that your ruler was properly calibrated and that you made your measurement correctly! If not then there could be even more uncertainty in your measurement.*Now let's say you want to calculate how much water there is in a swimming pool that is 20 m long and 15 m wide with a depth of 2 m.* Using these three dimensions we can calculate the volume…20m x 15m x 2m = 600m3*In this case we have reported our answer with 3 decimal places because that was how many were given in our original problem (i..e all numbers had 3 sig figs).* Notice how changing just one original value from 20m to 19m changes our final answer from 600m3to 570m3! This demonstrates why it's important for scientists and engineers use sig figs when performing calculations - one small change can yield vastly different results depending on which values have more or less precision.*It's also worth noting here that sometimes answers are given without any units attached… usually this happens when an exact value is known without any uncertainty i..e Avogadro's Number - 602 sextillion molecules!/A mole of anysubstance has a mass in grams equal to its atomic/molecular weight.How to Determine the Number of Significant FiguresThere are a few simple rules that you can follow to determine the number of significant figures in a given number:1) All non-zero digits are significant. This means that if a number is written as "2.345", then all four digits are significant because they are all non-zero.2) Zeros between non-zero digits are significant. This means that if a number iswritten as "2.0345", then all five digits are significant because the zero in between the 2 and the 3 is considered a non-zero digit.3) Leading zeros (zeros that come before the first non-zero digit) are not significant. This means that if a number is written as "0.0345", then only the three digits after the decimal point are considered significant because the leading zero is not considered a non-zero digit.4) Trailing zeros (zeros that come after the last non-zero digit but before a decimal point) may or may not be

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