Rounding. In mathematics rounding off is writing an answer to a given degree of accuracy.

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Dayna Barber
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1 Rounding In mathematics rounding off is writing an answer to a given degree of accuracy. Let's round off 314 to the nearest hundred. You know that 314 is closer to 300 than 400, so when we rounded off 314 to the nearest hundred is 300. Now let's round off 483 to the nearest hundred. We know that 483 is in between 400 and 500 and it is closer to 500. So, 483 rounded off to the nearest hundred is 500. This is a another way of looking at it: In 483, 4 is in Hundreds Place 8 is in Tens Place 3 is in Units Place Hundreds Tens Units or Ones We want to round off 483 to the nearest hundred.the number to the right of 4 is 8, which is more than 5.So you add 1 to the 4 ie 5 and change the digits in tens and units to zeros To round off a number correct to a given place, we round up (that is add 1) if the next figure is 5 or more, we round down (that is just drop them) if the next number is less than 5. eg: Round off 483 to the nearest hundred. 8 is more than 5 so we round up to 500. Round off 314 to the nearest hundred. 1 is less than 5, so you just drop14. That is we round down to 300.

2 Examples (a) Round off 483 to the nearest ten. Hundreds Tens Units or Ones In this number 8 represents tens. The number to the right of 8 is 3 which is less than 5, so we round down the number. (b) Round off to the nearest tenth. Tens Units Tenths Hundredths n this number 3 represents tenths. The number to the right of 3 is 2, which is less than 5, so just leave out to the nearest tenth is 67.3 Rounding off to the nearest tenth is sometimes referred to as rounding off to 1 decimal place. Rounding off to the nearest hundredths is sometimes referred to as rounding off to 2 decimal places. (c) Round off to 2 decimal places. (d) Round off $ to the nearest cent. Round off to the nearest cent means round off to the nearest hundredths or round off to two decimal places.

3 Many of the numbers we use represent situations which have directions as well as size The numbers which have a direction and a size are called directed numbers. Once a direction is chosen as positive (+), the opposite direction is taken as negative (- ). For example: If above zero degrees is positive (+), then below zero degrees is negative. If north is positive (+), then south is negative (-). If profit is positive (+), then loss is negative (-). Directed numbers are used in Mathematics, Engineering, Business and the Sciences. For example: -15, 8, 100, -100, -3.5, 0.33, are directed numbers. In the above example -15, 8, 100, -100 are called integers. When writing positive numbers you can leave the positive sign and just write the number. eg. +8 as 8 If a directed number is a whole number, it is called an integer. Example Addition of Directed Numbers Let's consider In this problem + and + signs are side by side.there is no number in between them. So the two positive signs which are side by side gives a positive sign. Remember this, Two like signs give a positive sign + + = = = 1 Sometimes directed numbers are written as Let's consider In this problem positive (+) and negative (-) signs are side by side without a number in between them. Two unlike signs which are side by side gives a negative (-) sign. Remember this:

4 Two unlike signs give a negative sign. + - = = -3-4 = - 7 Subtraction of Directed Numbers Let's consider In this problem the middle negative(-ve) signs are side by side without a number in between them. So the like signs which are side by side, always give a positive sign. This problem can also be written as = = 1-3 -(- 4) = = 1 and Let's consider In this problem negative(-ve) and positive(+ve) signs are side by side without a number in between them. That is two unlike signs are side by side, which gives a negative(-ve) sign. This problem can also be written as = -3-4 = - 7 and -3 - (+ 4) = -3-4 = -7 Multiplication of Directed Numbers Let's consider -3-4

5 When multiplying directed numbers Two like signs always give a positive(+ve) sign Two unlike signs always give a negative(-ve) sign (-ve) (-ve) = (+ve) -3-4 = = -12 (-ve) (+ve) = (-ve) Dividing directed numbers When dividing directed numbers Two like signs always give a positive(+ve) sign Two unlike signs always give a negative(-ve) sign Let's consider -3-4 Two like signs give a positive sign Let's look at this problem ( ) In this problem you can see all different operations, when we have more than one operation we have to follow the order of operations. ie. 1. Brackets 2. Division or multiplication from left to right 3. Addition or subtraction from left to right Let's do the brackets first ( ), inside this bracket you can see the multiplication and the subtraction signs. Remember the order, we have to do multiplication first and then the subtraction O.K 6-3 = - 18 (multiplying two unlike signs,gives a negative sign.) Then we subtract 2 ( ) = = -20 Now our problem -8 + ( ) = -8 + (-20) What's next?, addition,subtraction or division. Remember the order, division comes before addition and subtraction. O.K (-20) -4 = +5 (dividing two like signs, gives a positive sign.) Now our problem -8 + ( ) = -8 + (-20)

6 = (two like signs without a number in between them gives a positive sign) = now we are left with only addition and subtraction signs,so we can work out this problem from left to right. = = 5 It was stated in a newspaper that the attendance at the MCG(Melbourne Cricket Ground) for a football match was 64,000. But a friend who was attending the same match said the crowd was 64,492. The information from both sources are correct, but it was given to a different degree of accuracy. 64,492 might have been the exact number. When we round off 64,492 to two significant figures, it is 64,000. The first non-zero digit, reading from left to right in a number, is the first significant figure. e.g. In 64,492, 6 is the first significant figure.(sig.fig.) When we round off 64,492 to two sig. figs, that means in the answer we should have two non zero figures.the third figure(which is 4) is less than 5, so we drop them to zeros. Let's round off 64,492 to: (a) 1 significant figure which is 60,000 (b) 2 significant figures which is 64,000 (c) 3 significant figures which is 64,500 (d) 4 significant figures which is 64,490 (e) 5 significant figures which is 64,492 The accuracy of the answer will depend on the number of significant figures. The answer will be more accurate, if it is given to a higher number of significant figures. 64,492 is the most accurate answer and it is given to 5 sig. figs. *** The trailing zeros in a whole number are not significant.there are used to keep the other figures in there correct places. eg and 4 are significant not the zeros. *** The leading zeros in a decimal are not significant. There are used to keep the other figures in there correct places. eg , only 5 and 4 are significant. ***The zeros between the figures are significant. eg each figure is significant. There are 4 sig.figs. *** The last zero in a decimal is significant. eg. 3.20each figure is significant. There are 3sig.figs. eg. 0.50, 5 and last zero are significant.there are 2 sigfigs Examples

7 Example 1: Let's round off to: (a) 1 significant figure 9 is the first non-zero digit, that means 9 is the first sig. fig. In the second figure 2 which is less than 5, so we round down the number. When we round off to 1 sig. fig. is 90 (b) 3 significant figures In , 928 are the first three digits, the next figure 1 which is less than 5, so we round down the number. When we round off to 3 sig.figs. is 92.8 (c) 6 significant figures In , the first six significant numbers are The zeros between the figures are significant. the next figure 7 which is more than 5, so we round up the number. When we round off to 6sig. figs. is Example 2: Let's round off to: (a) 1 significant figure In , 4 is the first sig.fig. The leading zeros are not significant, but they are used to keep other figures in their correct places.in the above number the figure to the right of 4, is 6 which is more than 5, so we round up the number. When we round off to 1 sig.fig. is (b) 2 significant figures When we round off to 2 sig.figs. is (c) 4 significant figures In , 4 is the first sig.fig. The leading zeros are not significant, but they are used to keep other figures in their correct places.in the above number 5 is the 4th significant figure. The figure to the right of 5 is 3 which is less than 5, so we round down the number. When we round off to 4 sig.figs. is

8 Some numbers can be written in mathematical shorthand if the number is the product of "repeating numbers". eg 100 is the product of 10 multiplying itself two times: 100 = 10 x 10 or 64 is the product of 2 multiplying itself six times. 64 = 2 x 2 x 2 x 2 x 2 x 2 These numbers can be written in shorthand. and Index and base form The plural of "index" is "indices". Another name for index form is power form or power notation.

9 Index Law 1 Index Law 2 Index Law 3 ndex Law 4

10 Index Law 5 Index Law 6

13. [Place Value] units. The digit three places to the left of the decimal point is in the hundreds place. So 8 is in the hundreds column.

13. [Place Value] units. The digit three places to the left of the decimal point is in the hundreds place. So 8 is in the hundreds column. 13 [Place Value] Skill 131 Understanding and finding the place value of a digit in a number (1) Compare the position of the digit to the position of the decimal point Hint: There is a decimal point which