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## WHOLE NUMBERS AND FRACTIONS## Whole Numbers## The Decimal SystemOne of mans earliest needs was the development of a system for counting; that is, describing the number of objects in a group. Counting is taken for granted, so it is not obvious that at one time this was a difficult problem. Suppose there is a group of objects, like rocks. How is the number of rocks in the group described? The easiest method is to assign a symbol to represent the number. The number in each group can be determined by comparison with the number of fingers on the hand, and the appropriate symbol is then assigned. The problem becomes more complicated when the number of objects exceeds the number of fingers. For example, what symbol should be used to represent the number of objects below? Calling this *@#%! would quickly become very cumbersome. One of the great advances in mathematics was the development of the concept of There are ten separate symbols used, and these form the basis of the decimal system. These symbols are called The magnitude of a number is the sum of its digits, each multiplied by its place value (table below). Menu
## Other Base Number SystemsThe decimal, or base ten, system probably arose as a result of the fact that humans have ten fingers. There are other systems, however, that may be more suitable, depending upon the application. For example, a modern digital computer is really nothing more than a complex series of switches. Since there are only two possible positions for a switch, i.e., open or closed, it is more convenient for a computer to operate in a number system having only two digits. Such a system is called a This is a The decimal system equivalent of a binary number is the sum of each digit multiplied by its place value (table below). Therefore, the binary number 10110 represents the same number of objects as does the decimal number 22. In order to prevent confusion, a subscript that indicates the base usually is placed after the last digit. The binary number 10110 is written 101102, showing that the base is two, and the number is not interpreted as ten thousand, one hundred and ten. Strictly speaking, the decimal number 22 should be written as 2210, but since most of our work is in the decimal system, the base is understood to be ten and the subscript is omitted. ## AdditionHaving developed a system for representing the number of objects in a group, or counting, it is now necessary to define rules for combining two or more groups. This process is called This is the basic manner in which numbers are added. To shorten the process and to put it into a form that is convenient for calculation, the numbers may be tabulated vertically, taking care that the place values are properly aligned. The numbers to be added are called Note that when adding the 8 and the 3, a 1 is placed under the line in the units place and the 1 representing the number of groups of ten is ## Commutative LawThere are two laws that apply to addition. The first is thecommutative law, which states that two numbers may be added in either order and the result is the same. For example, 53 + 18 is equal to 18 + 53. The sum of either is 71. ## Associative LawThe second law is theassociative law, which states that addends may be combined in any order and the result is the same sum. This is shown below.So far, only numbers in the base ten system have been added, but addition can be performed in any number system. The rules are exactly the same. As an example, the binary numbers 1002 and 1112 will be added. The binary number 1002 represents one group of four objects plus no groups of two objects plus no single objects. The number 1112 represents one group of four objects plus one group of two objects plus one single object. The groups are added separately to obtain two groups of four objects plus one group of two objects plus one single object. However, two groups of four objects are the same as one group of eight objects. Thus, the total is one group of eight objects plus one group of two objects plus one single object, written as 10112. Notice that the unit in the ## SubtractionSubtraction is the process of determining the difference between two numbers or quantities, or the removal of some number of objects from a group. If the care is taken in writing the digits of the two numbers in neat columns, then the operation of subtraction is just a matter of subtracting the value of the digits in the second number from the corresponding place values in the first number. A minus sign (-) is used to indicate subtraction. Suppose there is a group of 53 objects from which 18 are to be removed. The 53 objects may be divided into 5 groups of 10 plus 3 single objects. This representation, however, is not convenient for removing 1 group of 10 plus 8 single objects. Instead, 53 objects are divided into 4 groups of 10 plus 13 single objects. After removing 1 group of 10 plus 8 single objects, there are 3 groups of 10 plus 5 single objects left, or the number 35.The number subtracted is called the Notice in this example that the digit in the units place of the minuend is less than the digit in the units place of the subtrahend. Because of this, it is necessary to As shown below, the commutative law does not apply to subtraction. where ≠ means Subtraction can be checked by adding the difference and the subtrahend; the sum will be the minuend. ## MultiplicationMultiplication is a short form of addition and is indicated by the times sign (x). The numbers multiplied are called themultiplier and the multiplicand. The result is called the product.The process of multiplication can be regarded as addition. In the above example, the number 3 is added 7 times; that is, 3 + 3 + 3 + 3 + 3 + 3 + 3. As shown below, multiplication obeys the commutative and associative laws. - Commutative Law: 3 x 7 = 7 x 3
- Associative Law: 2 x 3 x 5 = (2 x 3) x 5 = 2 x (3 x 5)
Since the commutative law applies to multiplication, the product 3 x 7 can mean: (1) add the number ## DivisionDivision is indicated by the division sign (Ã·). It is also indicated by placing one number over another separated by a horizontal line.The number to be divided is called the Division can be checked by multiplying the divisor and quotient. The result will be the dividend. Division is really a short form of subtraction. The above example started with a group of 28 objects, and 4 objects were removed from the group at a time. After this has been done 7 times, the group has been depleted; that is, none remain. Suppose instead, there is a group of 29 objects. If you are removing 4 objects at a time, after 7 times there will be 1 object left. This is called the The dividend is obtained by multiplying the divisor by the quotient and adding the remainder (4x7) + 1 = 29. ## Definition of FractionsWhole numbers are used for counting; that is, describing the number of objects in a group. However, the result of a measurement need not be a whole number, and in fact, rarely is. The number of pages in a book is, by definition, a whole number, but the width of a page in inches is not. We need a method the can describe the magnitude of numbers that lie between the whole numbers. This is achieved through the use of fractions.Suppose that we have a circle that has a total area of 12 square inches and we divide the circle into 4 equal parts. What will be the area of one of the parts? A fraction is the ratio of two whole numbers, and it indicates division. If the total area is divided by 4, the result will be the area of 1 of the parts. Total Area 12 square inches Each part of the circle is called a 1 = Numerator 4 = Denominator The meaning of a fraction, then, is that some entity is divided into an equal number of parts, indicated by the denominator, and then added as many times as indicated by the numerator. For example, what is meant by 9/16 of the area of a circle? To find out: - Divide the circle into 16 equal parts.
- Add 9 of these parts.
- The resulting area will be 9/16 of the area of the circle.
There are three different types of fractions. A The fractions 3/4, 7/16, and 191/327 are proper fractions; their values are less than 1. An The fractions 6/6, 16/7 and 23/13 are improper fractions; their values are equal to or greater than 1. A
## Adding and Subtracting FractionsIt is often necessary to add and subtract fractions. In adding and subtracting fractions, it is important to remember first what a fraction represents. It means a division of a unit into a number of equal parts, as indicated by the denominator, and then adding these parts as many times as indicated by the numerator. The fraction 2/7 means divide by 7 and add 2 times. The fraction 3/7 means divide by 7 and add 3 times. If these 2 fractions are added, the result will be the same as dividing by 7 and adding 5 times. The circle below is divided into 7 equal parts. Two of the parts are indicated by Xs. Three of the parts are indicated by 0s. The sum will be 5 of the parts, or 5/7 of the area of the circle. Fractions with the same denominator are added or subtracted by adding or subtracting their numerators and placing the result over the common denominator. For example: - Add 2/7 and 3/7.
- Subtract 7/11 from 9/11.
If the fractions to be added or subtracted do not have the same denominator, they cannot be added or subtracted directly as shown above. They must be altered so that they have the same denominator. Consider the fraction 3/4 and the circle shown below. If we divide the circle into four equal parts and add three of them, the result will be the area of three-quarters of the circle. On the other hand, if we divide the circle into 8 equal parts and add 6 of them, the result will be 6/8 of the area of the circle. Notice that this is exactly the same area as was represented by the fraction 3/4. Therefore, the fraction 6/8 is equivalent to the fraction 3/4. If the numerator and denominator of a fraction are multiplied by the same number, the value of the fraction is not changed. 3 x 2 = 6 4 x 2 = 8 This is the method that must be used when adding or subtracting fractions with different denominators. One of the fractions is altered to an equivalent form so that its denominator is the same as the denominator of the fraction to which it is to be added (or subtracted). The numerators are then added (or subtracted) and placed over the common denominator, as before. - Add 2/3 + 1/6.
- Note that 2/3 + 1/6 isNOTequal to:
A useful number in this process is the - What is 1/4 + 2/3 - 3/8?
- The lowest common denominator is 24.
When fractions are to be added or subtracted, their LCD must be found. ## Multiplying FractionsFractions are multiplied by multiplying the numerators together to obtain the numerator of the product, and multiplying the denominators together to obtain the denominator of the product. The following examples demonstrate that action- Multiply 2/3 and 5/7.
- Multiply 1/2 and 1/3.
Note that the numerators are multiplied directly, as are the denominators. There is no need to find a lowest common denominator, as is the case of addition and subtraction. The multiplication of 1/2 and 1/3 can be visualized with regard to the area of the circle. We first divide the area of the circle into two equal parts. This accounts for the fraction 1/2. This area is now divided into three equal parts. This accounts for the fraction 1/3. The final area is 1/6 the area of the circle. ## Dividing FractionsFractions may be divided by using the rule that the value of a fraction is unchanged if both the numerator and denominator are multiplied by the same number. The operation 1/2 divided by 1/3 is indicated as:If the numerator and denominator are multiplied by 3, we obtain: If we now multiply the numerator and denominator by 2, we obtain: This is the value of the quotient of 1/21/3. The same result is obtained simply by inverting the denominator and multiplying the result by the numerator. The process of inverting means to ## Reducing FractionsIn dealing with fractions, it is normal procedure to express the fraction in such a manner that the numerator and denominator are as small as possible. This is known as "reducing the fraction to its lowest terms." A fraction is said to be reduced to its lowest terms when the numerator and denominator have no common factor other than 1.Reducing a fraction to its lowest terms is made possible by the fact that the value of the fraction is unchanged ifboththe numerator and the denominator are divided by the same number. Although the fraction 6/8 is a perfectly proper fraction, it can be reduced to the fraction 3/4 by dividing the numerator and the denominator by 2. The fraction is now reduced to its lowest terms, since the numerator and denominator have no common factor other than 1. Another example would be to reduce the fraction 27/45 to its lowest terms. Finding the common factors for the purpose of fraction reduction normally is done by a trial-and-error procedure. ## Changing the Form of FractionsIn performing operations with fractions, it is sometimes necessary to alter them to an equivalent form for each type of computation. For example, if two mixed numbers, 2 1/3 and 7 1/8, must be multiplied, it becomes much easier if we change the mixed numbers to improper fractions first. A mixed number can be changed to an improper fraction by recognizing that a mixed number is just the sum of a whole number and a fraction.For example: Change the mixed number 4 2/3 to an improper fraction. Notice in this example that the LCD of the whole number and the fraction was found, and then both fractions were added. The same result is obtained, as shown below, by multiplying the whole number by the denominator of the fraction and then adding the numerator of the fraction to the result. The total is then placed over the denominator of the fraction. Two mixed numbers are multiplied together as follows: Multiply 2 1/3 by 7 1/8. Reduced to lowest terms. An improper fraction can be changed to a mixed number by dividing the numerator by the denominator, with any remainder placed over the denominator as the proper fraction. For example: Change the improper fraction 29/5 to a mixed number. |