WHOLE NUMBERS AND FRACTIONS
One 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 place value. The same symbol can be used to represent different numbers depending upon its position in the group of symbols that describes the number of objects. The symbol 1 standing alone means the group contains one object. The symbol 10 means the group contains ten objects. The symbol 1, then, has a different meaning depending upon its location. In this manner, a compact system for counting can be developed. A group containing two hundred and seventy-three objects can be described by the symbol 273, which means two groups of one-hundred plus seven groups of ten plus three groups of one.
There are ten separate symbols used, and these form the basis of the decimal system. These symbols are called digits, and the place values of the digits in the decimal system are multiples of ten, the base of the system (table below).
The magnitude of a number is the sum of its digits, each multiplied by its place value (table below).
The 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 binary system, and it contains only the digits 0 and 1.
This is a base two number system, and the place values are multiples of two, similar to the decimal system, in which the place values are multiples of ten (table below).
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.
Having 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 addition and is indicated by the plus sign (+). Addition can eliminate the need for counting the combined total and speeds up the process considerably. Suppose there is a group of fifty-three objects, represented by the decimal number 53. This means that there are five groups of ten objects plus three single objects. Another group of eighteen objects, represented by the decimal number 18, is added to this group. Thus, there is now an additional one group of ten objects plus eight single objects, making five groups of ten plus one group of ten for a total of six groups of ten. After combining the single objects, there are three single objects plus eight single objects for a total of eleven objects. However, eleven objects is the same as one group of ten plus one single object. Therefore, there is now a total of seven groups of ten objects plus one single object, which is represented by the number 71.
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 addends, and their total is the sum. Addition is simply a matter of combining the values of each number to arrive at the total value of the combined quantity. The use of place values for quantities greater than ten and the proper alignment of the digits before adding the numbers will help in coming up with the correct total. Note the previous example:
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 carried over to the tens place.
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 fours place was carried over to the eights place, just as when adding numbers in the decimal system. The addition can be checked by converting the numbers to the decimal system, as shown in the following example.
The number subtracted is called the subtrahend, the number from which the subtrahend is subtracted is called the minuend, and the result is the difference.
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 borrow 10 units, or reduce the tens place by 1, in order to subtract each group separately. This is shown in the following example:
As shown below, the commutative law does not apply to subtraction.
where ≠ means not equal to.
Subtraction can be checked by adding the difference and the subtrahend; the sum will be the minuend.
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.
Since the commutative law applies to multiplication, the product 3 x 7 can mean: (1) add the number 3 seven times, or (2) add the number 7 three times. Therefore, whether it is 7 groups of 3 objects or 3 groups of 7 objects, the result will always be a group of 21 objects.
The number to be divided is called the dividend; the number divided into the dividend is the divisor. The result is called the quotient. For example: 28 / 4 = 7, or:
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 remainder.
The dividend is obtained by multiplying the divisor by the quotient and adding the remainder (4x7) + 1 = 29.
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.
12 square inches
Each part of the circle is called a fourth and is denoted by the fraction 1/4. A fraction indicates division. The fraction 1/4 means 14.
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:
There are three different types of fractions. A proper fraction is one in which the numerator is less than the denominator, meaning it has a value less than 1.
The fractions 3/4, 7/16, and 191/327 are proper fractions; their values are less than 1.
An improper fraction is one in which the numerator is equal to or greater than the denominator. Its value will be equal to or greater than 1.
The fractions 6/6, 16/7 and 23/13 are improper fractions; their values are equal to or greater than 1.
A mixed number consists of a whole number and a fraction. The mixed number 3 means the whole number 3 plus the fraction . This is the type of number that would arise from the measurement of the width of this page. A likely result would be 8 inches plus inch, which equals 8 inches.
It 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:
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.
A useful number in this process is the lowest common denominator (LCD). The LCD is the smallest number that can be divided by all the denominators in a problem involving several fractions. In the above example, the number 6 is the LCD, since it is the smallest number that can be divided by both 6 and 3. The following is another example:
When fractions are to be added or subtracted, their LCD must be found.
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.
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 make the numerator the denominator and vice versa. Thus, if we invert 5/7, we obtain 7/5. Additional examples are as follows:
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.
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.