Introduction Prepared by Sa diyya Hendrickson Name: Date: Package Summary Defining Decimal Numbers Things to Remember Adding and Subtracting Decimals Multiplying Decimals Expressing Fractions as Decimals Shifting Decimal Points Irrational Numbers Let s Play! (Exercises) sadiyya.hendrickson@gmail.com 1 of 14 c Sa diyya Hendrickson
Decimal Numbers 1. decimal number: A decimal number is a rational number expressed in our Base 10 numeration system. A decimal point is used to separate the integer portion of the number (to the left) from the fractional portion (to the right). The positive and zero powers of 10 are represented by place values to the left of the decimal: 10 0 = ones place, 10 1 = tens place, 10 2 = hundreds place, etc. The negative powers of 10 are represented by place values to the right of the decimal: 10 1 = tenths place, 10 2 = hundredths place, etc. One strategy for remembering the order of the place values is to do a countdown:... 3, 2, 1, 0, 1, 2, 3,... Examples of Decimal Numbers 1. Every integer has a decimal representation. e.g. 38 = 38.0 2. The mixed number 567 1 has the decimal representation 567.25, where 567 is the 4 integer part and 0.25 represents the fractional part 1. 4 3. The decimal representation of a number may not be unique. For example, if the decimal part ends, we can add zeros at the end of the decimal part, and it remains the same! e.g. 38 = 38.00000 or 567.25 = 567.2500000 4. Consider the number 8.475624. What is the relationship between the number of places behind the decimal and the absolute value of the exponent for the last placevalue? sadiyya.hendrickson@gmail.com 2 of 14 c Sa diyya Hendrickson
Decimal Numbers Let s explore Example 2 in more detail: 1. Expanded Form: 567.25 = 5(10 2 ) + 6(10 1 ) + 7(10 0 ) + 2(10 1 ) + 5(10 2 ) ( ) ( ) 1 1 = 5(10 2 ) + 6(10) + 7(1) + 2 + 5 10 10 2 2. As a Fraction: (Finding an LCD) 567.25 = 56, 725 10 2 Question: Can you use your knowledge of LCDs and properties of exponents to justify how we arrived at the rational/fractional form? 567.25 = 5(10 2 ) + 6(10 1 ) + 7(10 0 ) + 2(10 1 ) + 5(10 2 ) = 5(102 ) 1 + 6(101 ) 1 + 7(1) ( ) ( ) 1 1 1 + 2 + 5 10 1 10 2 LCD = 5(102 )(10 2 ) + 6(101 )(10 2 ) + 7(1)(102 ) (1)(10 2 ) (1)(10 2 ) (1)(10 2 ) + 2(101 ) (10 1 )(10 1 ) + 5 10 2 = 5(102+2 ) 10 2 + 6(101+2 ) 10 2 + 7(102 ) 10 2 + 2(10) 10 1+1 + 5 10 2 = 5(104 ) 10 2 + 6(103 ) 10 2 + 7(102 ) 10 2 + 2(10) 10 2 + 5 10 2 = 5(104 ) + 6(10 3 ) + 7(10 2 ) + 2(10) + 5 10 2 = 56, 725 10 2 sadiyya.hendrickson@gmail.com 3 of 14 c Sa diyya Hendrickson
Things to Remember In the previous example, we noticed that the LCD of our fraction will always be determined by the place value of the last digit (since it will have the denominator with the largest power of 10). Recall that for integers, the last digit is in the ones place, and when we read the number, it tells us precisely how many ones we have! For example: 56 means 56 ones. This is true for decimal numbers, in general. For example: the last digit of 567.25 is 5 which is in the place, and we found that: ( ) 56, 725 1 567.25 = = (56, 725) 10 2 10 2 = 56, 725 hundredths! Exercise: Express 0.581 as a fraction. S1 Determine the last non-zero place-value: thousandths ( i.e. 10 3 = 1 10 3 ) S2 Write down number with the decimal point removed: (This tells us how many thousandths we have in all!) S3 Express the result in S2 numerically: (581) ( ) 1 10 = 581 3 10 3 thousandths Recall that in this number system, it s all about groups of 10! There are: ten hundredths in 10 1 since (10)(10 2 ) = 10 1 (ten hundredths in one tenth) ten tenths in 10 0 = 1, since (10)(10 1 ) = 10 1 1 = 10 0 = 1 ten ones in 10 1 = 10, since (10)(1) = 10 ten tens in = 100, since (10)(10) = 10 2 = 100... and so on When we exceed ten in any place-value category (e.g. hundredths, tenths, ones, tens, etc.), each group of ten can upgrade or be carried to the next place value! For example: If we have 23 tens, then 20 tens (i.e. two groups) can upgrade to 2 hundreds, leaving behind 3 tens, and no ones. This is why we write 23 tens as 23(10) = 230, which has 2 hundreds, three tens and zero ones. sadiyya.hendrickson@gmail.com 4 of 14 c Sa diyya Hendrickson
Adding and Subtracting To create numbers in the base 10 system, we must know the size of each place-value category. For instance, to build the number 23.45, we have to know that there are 2 tens, 3 ones, 4 tenths and 5 hundredths. Strategy When we are adding or subtracting numbers, we should be sure to align the place values so that we can easily count how many we have in each place-value category! Adding Decimals (e.g. 876.052 + 14.56) Subtracting Decimals (e.g. 876.052 14.56) sadiyya.hendrickson@gmail.com 5 of 14 c Sa diyya Hendrickson
Multiplying Decimals Suppose we were asked to calculate: 12 46 To do this without a calculator, many of us have learned the following approach: Let s use expanded forms and the distributive property to better understand this process. First we will recall that the distributive property requires the following: In words: If we are multiplying numbers such that some are in brackets and involve sums or differences, then everything in one pair of brackets must be multiplied with (i.e. distributed to) everything on the outside of those brackets. Some generalizations of the property include: Now consider the product expressed using expanded forms: 12 46 = (2 + 10)(6 + 4(10)) Can you use the distributive property to reach the solution of 552? Be sure to keep up with how many ones, tens and hundreds you have and the need for upgrades! 12 46 = (2 + 10)(6 + 40) = (2 + 10)(6 + 4(10)) =. sadiyya.hendrickson@gmail.com 6 of 14 c Sa diyya Hendrickson
Multiplying Decimals Below is a detailed solution of the exercise on the previous page: 12 46 = (2 + 10)(6 + 40) (1) = (2 + 10)(6 + 4(10)) (2) = 2(6) + 2(4(10)) + 10(6 + 4(10)) distributing the 2 (3) = 12 + 8(10) + 10(6 + 4(10)) 12 ones needs an upgrade! (4) = 2 + 10 + 8(10) + 10(6 + 4(10)) (5) = 2 + 9(10) + 10(6 + 4(10)) (6) = 2 + 9(10) + (10)6 + 10(4(10)) distributing the 10 (7) = 2 + 9(10) + 6(10) + 4(10 2 ) (8) = 2 + 15(10) + 4(10 2 ) 15 tens needs an upgrade! (9) = 2 + 5(10) + 10(10) + 4(10 2 ) (10) = 2 + 5(10) + 5(10 2 ) (11) = 552 (12) In the diagram below, fill in the number of the highlighted line (above) that corresponds to the detail that has been pointed out. Two have been completed, so be sure to understand those answers before you begin! Line 5 is upgrading the group of 10 ones in 12 (i.e. we are carrying one ten). Line 8 shows the result of distributing 10, which produces the second highlighted row in the diagram equaling 460 = 4(10 2 ) + 6(10)! Note: The 6 was written in the tens place, underneath the 9 because it was created by multiplying a 1 in the tens place (i.e. one ten) with six ones, which creates six tens. sadiyya.hendrickson@gmail.com 7 of 14 c Sa diyya Hendrickson
Multiplying Decimals Calculate: (23.48)(3.24) Solution: A popular approach to multiplying decimals is to: S1 Determine the product of the numbers when the decimals are removed: i.e. 2348 324 S2 Determine the number of digits needed behind the decimal point in the solution by adding up the total number of digits behind each of the two decimals. There are two digits behind each decimal, giving a total of 2 + 2 = 4 decimal places needed in the solution. 7 6. 0 7 5 2 }{{} 4 places Why Does This Work? Consider the fractions approach: We know that: 23.48 = 2348 10 2 and 3.24 = 324 10 2 Therefore, the product is given by: 2348 10 324 2348 324 = 2 102 10 4 1. On page 3, we discovered that the numerators of these fractions will always be the number with the decimal point removed, which is what S1 suggests that you use. 2. We also know that the power of 10 in the denominator tells us how many places there are behind the decimal point. By properties of exponents, this will always be the sum of the powers of 10 in each decimal number s denominator (i.e. the sum of the number of places behind each decimal). sadiyya.hendrickson@gmail.com 8 of 14 c Sa diyya Hendrickson
Fractions to Decimals Two Kinds of Repeating Decimal Numbers: 1. Repeating, Non-Terminating Decimal Representations: A decimal representation that has a string of digits (other than zero) that repeats. (The first string of repeating digits is called the repetend). For example, 0.1234545... = 0.12345 has the repetend 45. The convention is to write a bar over the repetend instead of writing ellipses. 2. Repeating, Terminating Decimal Representations: A decimal representation with a finite decimal expansion. The repetend for these numbers is zero, which is not usually written. These numbers occur for fractions who have equivalent fractions with powers of 10. e.g. 0.123 = 0.1230. Celebrity Terminating Decimals: There are some very popular terminating decimals whose fractions we d want to know! 1 4 = 0.25 1 2 = 0.5 3 4 = 0.75 1 5 = 0.2 Notice that it s fairly easy to see the smallest power of 10 that we can create from their denominators! Can you identify them? For example, what is the smallest power of 10 that has 4 as a factor? Fractions whose Denominators are Powers of 10 If we get a fraction whose denominator is 10, 100, 1000, 10000, etc, our work is already done! These are all powers of 10. The number of zeros tells us the power! e.g. 47 1000 = 7 10 3 We have 47 thousandths. The denominator tells us that the decimal representation has three digits behind the decimal point. We also know that the 7 must be the digit in the thousandths place. So we work backwards, filling in the blanks from right to left, filling in empty spaces with zeros:. 4 7 sadiyya.hendrickson@gmail.com 9 of 14 c Sa diyya Hendrickson
Fractions to Decimals The general method for finding the decimal representation of a fraction is long division. Let s look at an example and recall our strategies. Exercise 1: Express 1 11 as a decimal. Solution: Notice that 100 has appeared again, meaning that a pattern has begun and our repetend is 09. Therefore: 1 11 = 0.09 Exercise 2: Express 6 7 as a decimal. A Few Questions: 1. How many remainders of 7 did you see before the decimal expansion started to repeat? 2. What is the total number of possible remainders when dividing by 7? 3. Is it possible for us to perform the long division indefinitely? sadiyya.hendrickson@gmail.com 10 of 14 c Sa diyya Hendrickson
Shifting Decimal Points Consider taking a decimal number and multiplying it by powers of 10. What do you think might happen to the number? Let s explore an example using expanded form and the distributive property: Multiplying by 10 2 simply caused our decimal to shift places to the. Can you make a prediction of what will happen when we divide by powers of 10? Dividing by 10 4 caused our decimal to shift places to the. sadiyya.hendrickson@gmail.com 11 of 14 c Sa diyya Hendrickson
Power of Fractions The Power of Fractions Sometimes operations with decimals are more cumbersome and tedious then with fractions. So, working towards being comfortable with both fractions and decimals will help you move through mathematics with more ease. Consider the following exercise: 0.75 2.25 1. Option 1: Use fractions 0.75 2.25 = 3 4 21 4 rewriting decimals as fractions = 3 4 9 4 = 3 4 4 9 = 1 1 1 3 changing mixed number to improper fraction by theorem for fraction division by reducible pairs (3, 9) and (4, 4) = 1 3 = 0.3 2. Option 2: Use long division Notice that using fraction notation, our problem is: 0.75 2.25. If we wanted to create an equivalent fraction so that our denominator is a whole number, what should we multiply the numerator and denominator by so that our decimal moves over two places to the right? 0.75 2.25 = (0.75)( ) (2.25)( ) = 75 225 225 ) 75 This is a case of a question that is much simpler with fractions, simply because you can avoid having to work with large numbers. In general, it s a good idea to keep fractions in mind when you re working with decimals! sadiyya.hendrickson@gmail.com 12 of 14 c Sa diyya Hendrickson
Irrationals decimal and fraction representations: Every fraction has a repeating decimal representation and every repeating decimal representation has a fraction representation. Q: Can every number on the number line be expressed as a fraction? A: No There are infinitely many numbers that cannot be expressed in the form of a fraction. These numbers are not rational and consequently are called irrational. They are also characterized by their non-repeating, non-terminating decimal expansions. In other words, their decimal expansions never repeat making it impossible for us to express them numerically. Symbols are used to represent them since they don t have a closed numerical representation. Celebrity Irrational Numbers: Very well known irrational numbers include: 1. π = 3.141592654... the ratio of a circles circumference and diameter. 2. 2 = 1.41421356237... This number appears as the length of a diagonal in a square with side lengths equal to 1. 3. e = 2.718281828459045... This number is named after a very well-known mathematician named Euler, who contributed greatly to the development of calculus, among many other things. sadiyya.hendrickson@gmail.com 13 of 14 c Sa diyya Hendrickson
Let s Play! 1. Express the following decimals as mixed number or fractions in reduced form: (a) 0.05 (b) 12.36 (c) 0.763 (d) 120.5 (e) 45.25 (f) 1.498 2. Express the following in decimal form: (a) four-hundred fifty-six tenths (b) two-thousand six-hundred and eight ten-thousandths (c) 5(10 3 ) + 3(10 2 ) + 9(10) + 10 0 + 4(10 1 ) (d) 7(10 6 ) + 6(10 4 ) + 5(10 3 ) + 10 2 + 8(10 2 ) 3. Calculate: (a) 45.16 + 47.325 (b) 40.563 32.981 (c) 6.542 + 0.798 (d) 364.2 273.16 (e) 53.7 + 42.0513 (f) 2.398 0.099 4. Calculate the following products using fractions: (a) 42 0.5 (b) 4 0.75 (c) 0.125 0.16 (d) 0.05 40 (e) 0.24 50 (f) 5.6 0.625 5. Calculate the following products: (a) 476.2 52.1 (b) 36.516 0.21 (c) 45.63 0.59 (d) 407.1 36.54 6. Calculate the following products: (a) 3.45 10 3 (b) 476.3215 10 2 (c) 542.3876 10 4 (d) 36.5 10 5 7. Express the following fractions as decimals: (a) 25 (b) 4563 (c) 30 (d) 5648348 10 6 10 2 10 3 10 5 (e) 1 6 (f) 5 8 (g) 3 5 (h) 13 9 sadiyya.hendrickson@gmail.com 14 of 14 c Sa diyya Hendrickson