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# Decimals

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### Decimals

1. 1. Definition of DecimalThe word "Decimal" really means "basedon 10" (From Latin decima: a tenth part).We sometimes say "decimal" when wemean anything to do with our numberingsystem, but a "Decimal Number" usuallymeans there is a Decimal Point.
2. 2. PLACE VALUETo understand decimals wemust know about PLACEVALUEWhen we write numbers,the position (or "place") ofeach number is important.In the number 327:The "7" is in the Unitsposition, meaning just 7 (or7 "1"s),the "2" is in the Tensposition meaning 2 tens (ortwenty), "Three Hundred Twentyand the "3" is in theHundreds position, meaning Seven"3 hundreds.
3. 3. As we move left, each position is 10 times bigger!From Units, to Tens, to Hundreds ... and ... As we move right, each position is 10 times smaller. From Hundreds, to Tens, to Units
4. 4. What is 10 times smallerthan Units?Ans.1/ ths (Tenths) 10 But we must first write a decimal point, so we know exactly where the Units position is "three hundred twenty seven and four tenths" And that is a Decimal Number!
5. 5. DECIMAL POINTThe decimal point is the most important part of a Decimal NumIt is exactly to the right of the Units position.Now we can continue with smaller and smallervalues, from tenths, to hundredths, and so on, likein this example:
6. 6. LARGE & SMALLSo, our Decimal System lets us write numbersas large or as small as we want, using thedecimal point.Numbers can be placed to the left or right of adecimal point, to indicate valuesgreater than one or less than one.
7. 7. The number to the left of the decimal point is a whole number (17 for example)As we move further left, every number place gets 10times bigger. The first digit on the right means tenths (1/10). As we move further right, every number place gets 10 times smaller (one tenth as big).
8. 8. Youacould think of a Hundredths, etc ... as Whole Number Plus Tenths,decimal number as a wholenumber plus tenths,hundredths, etc:Example 1: What is 2.3 ? On the left side is "2", that is the whoThe 3 is in the "tenths"position, meaning "3 tenths", or3/10So, 2.3 is "2 and 3 tenths"
9. 9. ... as a Decimal FractionYou could think of a decimal number as a Decimal Fraction.A Decimal Fraction is a fraction where thedenominator (the bottom number) is a number such as 10, 100, 1000, etc (in otherwords a power of ten)So "2.3" would look like this: 23/10And "13.76" would look like this: 1376/100
10. 10. ... as a Whole Number and Decimal FractionYou could think of a decimal number as a Whole Numberplus a Decimal Fraction.So"2.3" would look like this: 2 & 3/10And"13.76" would look like this: 13 & 76/100
11. 11. What is "Rounding" ?Rounding means reducing the digits in a number whiletrying to keep its value similar.The result is less accurate, but easier to use.Example: 73 rounded to the nearest ten is 70,because 73 is closer to 70 than to 80.
12. 12. How to Round Numbers Decide which is the last digit to Leave it the same if the next digit is less than keep (this is called rounding down) But increase it by 1 if the next digit is 5 or more (this is called rounding upto the neare Example: Round 74 ) to keep the "7" as it is in the 10s positiont is "4" which is less than 5, so no change is needed Answer: 70 (74 gets "rounded down")
13. 13. First you need to know if you are rounding to tenths,Or maybe to "so many decimal places".That tells you how much of the number will be left wh Examples Because ...3.1416 rounded to ... the next digit (1) ishundredths is 3.14 less than 51.2635 rounded to ... the next digit (6) istenths is 1.3 5 or more1.2635 rounded to 3 ... the next digit (5) isdecimal places is 1.264 5 or more
14. 14. You may want to round to tens, hundreds, etc, In this case you replace the removed digits with zero. Examples… Because ... ... the next digit (4) is less than134.9 rounded to tens is 130 512,690 rounded to thousands is ... the next digit (6) is 5 or13,000 more ... the next digit (2) is less than1.239 rounded to units is 1 5
15. 15. Example: Add 3.25, 0.075 and 5 Line the decimals up: 3.25 0.075 + 5. "Pad" with zeros: 3.250 0.075 + 5.000 Add: 3.250 0.075 + 5.000 8.325
16. 16. Subtracting decimals is easy if you keep your work neat!To subtract decimals, follow these steps:
17. 17. Example: Calculate 7.005-0.55Line the decimals up: 7.005 - 0.55"Pad" with zeros: 7.005 - 0.550Subtract: 7.005 - 0.550 6.455
18. 18. Multiply without the decimal point, then re-insert it in the correct spot! To multiply decimals follow these steps:tiply normally, ignoring the decimal points. Then put the decimal point in the answer - it will have as many In other places as the count up decimal words, just two original numbers combined. are after how many numbers the decimal point in both numbers you are multiplying, then the answer should have that many numbers after its decimal point.
19. 19. Example: Multiply 0.03 by 1.1start with: 0.03 1.1multiply without decimal points: 3 11 = 330.03 has 2 decimal places,and 1.1 has 1 decimal place,so the answer has 3 decimal places: 0.033
20. 20. Quick method: use Long Division without the decimal point, then re-insert the decimal point in the answer.To divide a decimal number by a whole number: Use Long Division (ignoring the decimal point) Then put the decimal point in the samespot as the dividend (the number being divided)
21. 21. Example: Divide 9.1 by 7Ignore the decimal point and use Long Division: 13 7 )91 9 7 21 21Put the decimal 0point in theanswer directly above thedecimal point in the dividend: 1.3 7 )9.1 The answer is 1.3
22. 22. The trick is to convert the number you are dividing by to a wholenumber first, by shifting the decimal point of both numbers to theright:Now is safe to do this if number, and can continIt you are dividing by a whole youremember to shift the decimalpoint of both numbers the samenumber of places.
23. 23. 1.1You are not dividing by a move 1whole number, so you need to 5.39 1.1 53.9 11move the decimal point so move 1that you are dividing by aYou are now dividing by a 049whole number: 11 )539whole number, so you can 5 0proceed: 53Ignore the decimal point and 44 99use Long Division: 99 0Put the decimal point in the 04.9answer directly above the 11 )53.9decimal point in the dividend: The answer is 4.9
24. 24. To convert a Decimal to a Fraction followStep 1: Write down the decimaldivided by 1.Step 2: Multiply both top andbottom by 10 for every numberafter the decimal point. (Forexample, if there are twonumbers after the decimal, thenuse 100, if there are three then
25. 25. Example 1: Express 0.75 as a fraction 0.75Step 1: Write down: 1Step 2: Multiply both top and bottomby 100 (because there were 2 digits 100after the decimal place): 0.75 75 = 1 100 100
26. 26. 25 Step 3: Simplify the fraction: 75 3 = 100 4 2500 is called a decimal fraction and 3/4 is called a commo
27. 27. Example 2: Express 0.333 as a fraction 0.333Step 1: Write down: 1Step 2: Multiply both top and bottom by1,000 (there were 3 digits after the decimalplace so that is 10 10 10=1,000) 333 1,000Step 3: Simplify Fraction: any simpler! Cant get Answer = 333/1,000
28. 28. But a Special Note:If you really meant 0.333... (in other words 3s repeating forever whichis called 3 recurring) then we need to follow a special argument. Inthis case we would write down: 0.333... 1 3 0.333... 0.999...Then MULTIPLY both sides by 3: = 1 3And 0.999... = 1 so: 3 Answer = 1/3
29. 29. Convert Fractions to DecimalsTo convert a Fraction to a Decimal , follow theStep 1: Find a number you can multiply by thebottom of the fraction to make it 10, or 100, or1000, or any 1followed by 0s.Step 2: Multiply both top and bottom by thatnumber.Step 3. Then write down just the top number,putting the decimal place in the correct spot (one
30. 30. Example 1: Express 3/4 as a DecimalStep 1: We can multiply 4 by 25 to become 100Step 2: Multiply top and bottom by 25: 25 3 75 = 4 100 25Step 3: Write down 75 with the decimal place 2 spacesfrom the right (because 100 has 2 zeros); Answer = 0.75
31. 31. Example 2: Express 1/3 as a Decimal Step 1: There is no way to multiply 3 to become 10 or 100 or any "1 followed by zeros", but we can calculate an approximate decimal by choosing to multiply by, say, 333 333 Step 2: Multiply top and bottom by 333: 1 333 = 3 999 333 Step 3: Now, 999 is nearly 1,000, so let us write down 333 with the decimal place 3 spaces from the right (because 1,000 has 3 zeros):wer = 0.333 (accurate to only 3 decimal plac
32. 32. Convert Decimal to PercentJust move the decimal point 2 places to the right and add a "%" sign! The easiest way to multiply by 100 is to move the decimal point 2 places to the right: To Percent From Decimal move the decimal point 2 places to the right, and add the "%" sign.
33. 33. Convert Percent to Decimal Just move the decimal point 2 places to the left and remove the "%" sign!The easiest way to divide by 100 is to move the decimal point2 places to the left.So: To DecimalFrom Percent move the decimal point 2 places to the left, and remove the "%" sign.
34. 34. THEREFORE,FROM THE PRESENTATION WECOME TO KNOW THAT DECIMALS IS A VERYEASY TOPIC AND IS CLOSELY RELATED TOFRACTIONS & PERCENTAGES. MOREOVER,DECIMAL NO.S ARE OF MUCH SIGNIFICANCE IN MATHEMATICS & OUR DAILY LIFE.