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3.1: Required Math Concepts

  • Page ID
  • Numeral System and Notation


    Natural numbers

    Whole numbers

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


    \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 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.

    \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.

    \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.

    Improper and Proper Fractions

    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.

    \frac{1}{2} , \frac{3}{7} , and \frac{11}{18} are proper fractions.

    Equivalent 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.

    \frac{1}{2} is equivalent to \frac{4}{8}.

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