A store front display in NYC showing price tags with decimals. The Use of Decimals in the Real World

Place Values in a Decimal and the Expanded Form of a Decimal. Given a decimal place value is is valueplace is valueplace is valueplace 10 1 is valueplace

Place Values in a Decimal and the Expanded Form of a Decimal. Given a decimal We can therefore rewrite it in an expanded form which can then be converted to a mixed number 

How a decimal is read? A decimal is read as if they were written in fraction form except that the decimal point is read “and”. We don’t use the word “and” in any other places. Example: is read as “One thousandtwo hundredfourandsix hundred fifty seventhousandths” i.e. This method works only for short decimals, and when there are many digits after the decimal point, such as , the scientists and engineers will call it “Two point seven one eight two eight”

How is a decimal read? A problem for discussion Is there any chance of confusion when a student reads the decimal as “One thousand and two hundred and four and thirty eight hundredths” ? Response In daily uses, it seldomly causes a problem, but it doesn’t mean that confusion will never happen either. For instance, consider the number 36, If you read this as “Thirty six thousand and eight hundred and forty one thousandths”, then the person who heard this may interpret it as “ ”

Remarks 1. The decimal notation is not unique throughout the world even up to this day. For example, the British uses 3·1416 (with the dot higher) while the French and German use 3, When we change a fraction into a decimal, the representation is not always terminating. and we will see later that even terminating decimals have non-terminating representations such as 0.25 = ….  eg.

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Comparing Decimals Research shows that most students believe that is bigger than 0.35 because a)0.287 has more digits than 0.35 b)0.287 is read as “two hundred eighty seven thousandths” which sounds larger than “thirty five hundredths”, particularly when they are not familiar with the fact than “one thousandth” is really smaller than “one hundredth”.

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 =

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 =

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 =

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 =

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 =

Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg × 10 3 Dividing a decimal by 10 n is the same as moving the decimal point to the left n places, adding place holders if necessary. eg × 10 5 = eg ÷ 10 3 = eg ÷ 10 2 = =

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up. (click to see the animation)

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal.

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point.

Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point.

Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up

In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. (Please click to see animation.) Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up

In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up

In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, or ? Solution We first put one of them above the other such that the decimal points are lined up Now we see that the upper digit in the highlighted column is larger, there for the corresponding number (i.e. the upper one) is larger than the lower one.

Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers, except that the numbers are aligned by the decimal points rather than the last digits (counting from the left). Example ────── Incorrect ─────── Correct, and we have to treat any empty space as a 0.

There are two ways to carry out this operation, (I)Converting the decimals to fractions Example: 0.65 × Multiplication of Decimals =      000, 

(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers. In the end we insert the decimal point back to the answer in the proper position. Example: 0.65 × can be first treated as 65 × 2417 = , the decimal point is then re-inserted to the product such that “the number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicands.” In this particular case, 0.65 has two decimal places and has three decimal places. Hence their product should have 5 decimal places, and this means that 0.65 × =

Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. (click)

Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. (click)

Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount.

Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. (click)

Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend.

Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. (click) Division of Decimals

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers

Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: ÷ 1.2 Step 4. Divide as if we are dividing whole numbers Therefore ÷ 1.2 = 2.13

Converting Fractions to Decimals Exploration: Find the first 3 digits in the decimal expansion of 4 / 7. We first consider the following From long division we have 4000 ÷ 7 = 571 r3, hence It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page) 

Converting Fractions to Decimals Conclusion The decimal expansion of a fraction a/b can be obtained by long division ···

Converting Fractions to Decimals Fact: The decimal expansion of any fraction a/b is either terminating or repeating. Theorem: If the fraction a/b is in its reduced form, then its decimal expansion is terminating if and only if b is one of the following forms. (1) a product of 2’s only, (2) a product of 5’s only (3) a product of 2’s and 5’s only. Examples: is not terminating 11 3 is terminating is terminating 

Converting Fractions to Decimals Theorem: If the fraction a/b is in its reduced form, and b = 2 m 5 n then the decimal expansion of a/b is terminating with number of decimal places exactly equal to max{m, n} Now we know what kind of fractions will have terminating decimal expansions, but can we predict how many decimal places there will be in the expansion? Example: The decimal expansion of  will have exactly 3 decimal places.

Converting Fractions to Decimals One more question: If we know that a certain fraction has repeating decimal expansion, can we predict its cycle length? Unfortunately there is no formula to calculate the precise cycle length. All we know is an upper bound and a small (not too helpful) property. Theorem If p is a prime number other than 2 and 5, then the cycle length of 1/p is at most (p – 1), and the cycle length must divide (p – 1). Example: The cycle length of 1/31 is at most 30, and it must divide 30. In fact, the cycle length of 1/31 is 15.

Converting Fractions to Decimals More examples prime number pcycle length of 1/p There is no obvious pattern on the cycle length, and a large denominator can have a small cycle length.

Converting Fractions to Decimals More facts (optional) 1.If p is a prime other than 2 or 5, then the cycle length of 1/(p 2 ) is at most p(p – 1) and the cycle length must divide p(p – 1). 2. If p and q are different primes other than 2 and 5, then the cycle length of 1/pq will be at most (p – 1)(q – 1) and divides (p – 1)(q – 1). Example: cycle length of 1/7 is 6, cycle length of 1/49 is 42 (= 7×6). Example: Cycle length of 1/(7×11) is less than 6×10 = 60, and must divide 60. It turns out that the cycle length of 1/77 is only 6.

Converting Decimals to Fractions From the previous theorem, we see that only repeating or terminating decimals can be converted to a fraction. Procedures: (1)terminating decimal, eg = / The number of 0’s in the denominator is equal to the number of decimal places. (2)repeating decimals of type I, eg ··· = 2 / ··· = 47 / ··· = 528 / 999

Converting Decimals to Fractions Procedures: 2)repeating decimals of type I, eg ··· = 2 / ··· = 47 / ··· = 528 / 999 3)repeating decimals of type II, eg ··· = 62 / ··· = 62 / ··· = 3449 /

Converting Decimals to Fractions Procedures: 3)repeating decimals of type II, eg ··· = 62 / ··· = 62 / ··· = 3449 / ) repeating decimals of type III, eg ··· = ···

Converting Decimals to Fractions Procedures: 3)repeating decimals of type II, eg ··· = 62 / ··· = 62 / ··· = 3449 / ) repeating decimals of type III, eg ··· = ···