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3.1: Required Math Concepts

  • Page ID
    2316
  • Numeral System and Notation

    It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

    Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

    But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations.

    Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

    Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

    Numbers

    Natural numbers

    The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1,2,3,…} where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers.

    Whole numbers

    If we add zero to the counting numbers, we get the set of whole numbers.

    • Counting Numbers: 1, 2, 3, …
    • Whole Numbers: 0, 1, 2, 3, …

    Integers

    A set of integers adds the opposites of the natural numbers to the set of whole numbers:{,−3,−2,−1,0,1,2,3,}. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

    \begin{array}{lllll}\stackrel{\text{negative integers}}{\stackrel{}{\dots ,-3,-2,-1,}}\hfill & \hfill & \stackrel{\text{zero}}{\stackrel{}{0,}}\hfill & \hfill & \stackrel{\text{positive integers}}{\stackrel{}{1,2,3,\cdots }}\hfill \end{array}

    Fractions

    Often in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of sugar. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. These people need to use numbers which are part of a whole. These numbers are very useful both in algebra and in everyday life and they are called factions. Hence, Fractions are a way to represent parts of a whole. It is written \frac{a}{b}, where a and b are integers and b\ne 0. In a fraction, a is called the numerator and b is called the denominator. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined.

    In (Figure), the circle has been divided into three parts of equal size. Each part represents \frac{1}{3} of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

    A circle is divided into three equal wedges. Each piece is labeled as one third.
    Doing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators.

    What does the fraction \frac{2}{3} represent? The fraction \frac{2}{3} means two of three equal parts.

    A circle is divided into three equal wedges. Two of the wedges are shaded.

    An interactive or media element has been excluded from this version of the text. You can view it online here: http://pressbooks.oer.hawaii.edu/buildingmaint/?p=142

    Improper and Proper Fractions

    In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. Fractions such as \frac{5}{4} , \frac{3}{2} , \frac{5}{5} , and \frac{7}{3} are called improper fractions.

    When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as \frac{1}{2} , \frac{3}{7} , and \frac{11}{18} are proper fractions.

    Equivalent Fractions

    Equivalent fractions are fractions that have the same value. For example, the fractions \frac{6}{6} and \frac{8}{8} have the same value, 1. Figure shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that \frac{1}{2}is equivalent to \frac{4}{8} . In other words, they are equivalent fractions.

    Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.

    Figure #. Since the same amount is of each pizza is shaded, we see that \frac{1}{2} is equivalent to \frac{4}{8}.

    Add or Subtract Fractions

    To add or subtract fractions, they must have a common denominator. If the factions have the same denominator, we just add the numerators and place the sum over the common denominator. If the fractions have different denominators, what do we need to do?

    First, we will use fraction tiles to model finding the common denominator of \frac{1}{2} and \frac{1}{3}.

    We’ll start with one \frac{1}{2} tile and \frac{1}{3} tile. We want to find a common fraction tile that we can use to match both\frac{1}{2} and \frac{1}{3} exactly.

    If we try the \frac{1}{4} pieces, 2 of them exactly match the \frac{1}{2} piece, but they do not exactly match the \frac{1}{3} piece.

    Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.

    If we try the \frac{1}{5} pieces, they do not exactly cover the \frac{1}{2} piece or the \frac{1}{3} piece.

    Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.

    If we try the \frac{1}{6} pieces, we see that exactly 3 of them cover the \frac{1}{2} piece, and exactly 2 of them cover the \frac{1}{3} piece.

    Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle are three smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 half rectangle. Below the 1 third rectangle are two smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 third rectangle.

    If we were to try the \frac{1}{12} pieces, they would also work.

    Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.

    Even smaller tiles, such as \frac{1}{24} and \frac{1}{48}, would also exactly cover the \frac{1}{2} piece and the \frac{1}{3} piece.

    The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of \frac{1}{2} and \frac{1}{3} is 6.

    Notice that all of the tiles that cover \frac{1}{2} and \frac{1}{3} have something in common: Their denominators are common multiples of 2 and 3, the denominators of \frac{1}{2} and \frac{1}{3}. The least common multiple (LCM) of the denominators is 6, and so we say that 6 is the least common denominator (LCD) of the fractions \frac{1}{2} and \frac{1}{3}. Therefore, the least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

    To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.

    Find the LCD for the fractions \frac{7}{12} and \frac{5}{18}.

    Solution
    Factor each denominator into its primes. .
    List the primes of 12 and the primes of 18 lining them up in columns when possible. .
    Bring down the columns. .
    Multiply the factors. The product is the LCM. \text{LCM}=36
    The LCM of 12 and 18 is 36, so the LCD of \frac{7}{12} and \frac{5}{18} is 36. LCD of \frac{7}{12} and \frac{5}{18} is 36.