{"id":1130,"date":"2026-05-12T21:46:19","date_gmt":"2026-05-12T21:46:19","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/equations-a-general-discussion\/"},"modified":"2026-05-13T03:10:25","modified_gmt":"2026-05-13T03:10:25","slug":"equations-a-general-discussion","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/equations-a-general-discussion\/","title":{"raw":"4.2 Equations: A General Discussion","rendered":"4.2 Equations: A General Discussion"},"content":{"raw":"
\r\n
\r\n\r\nFor someone not used to dealing with equations and mathematical statements, the key to understanding them is to think about the physical and conceptual relationships that the equation expresses. Let\u2019s take a famous equation\u2014shown in Equation\u00a0(32)<\/span><\/a>\u2014and think about what it means conceptually.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]E = {mc}^{2}[\/latex]<\/th>\r\n(32)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nWhere E represents energy, [latex]m[\/latex] represents mass, and [latex]c^{2}[\/latex] is a very large number: the square of the speed of light. What this equation means, in Einstein\u2019s words (Mayne, 1948), is:\r\n\r\n<\/div>\r\n
\r\n\r\n\u201cIt followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing\u2014a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c\u2011squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before.\u201d\r\n\r\n<\/div>\r\n
\r\n\r\nThe point of the equation is that matter (i.e., mass) and energy are fundamentally the same thing. The equation tells us that energy is equal to mass times a very large number ([3.0\u00a0x\u00a0108<\/sup> m\/s]2<\/sup>), indicating that a small amount of mass is equal to a large amount of energy. This is a very significant concept, especially to the residents of Hiroshima and Nagasaki, Japan, during the mid\u20111940s (where 64 kg of enriched uranium was converted to 16 kilotons of energy).\r\n\r\n<\/div>\r\n
\r\n\r\nA useful approach to understanding equations and mathematics is to look at the relationships between the various parts of the equation and determine how those parts relate to reality. Here are basic steps for that approach.\r\n\r\n<\/div>\r\n
    \r\n \t
  1. Look at each item in the equation and determine what it represents (e.g., length; density; velocity). It also helps to figure out the dimensions of each variable (e.g., length, L; density ML-3<\/sup>; velocity, LT-1<\/sup>) and their associated units (e.g., meters; kg\/m3<\/sup>; m\/s).<\/li>\r\n \t
  2. Determine if each item varies or is constant. Constants are items with a value that does not change such as gravitational acceleration on Earth, the speed of light in a vacuum, and the number of molecules in a mole of a substance.<\/li>\r\n<\/ol>\r\n
      \r\n \t
    1. Compare the variables on one side of the equation to those on the other side. Look for numerator\u2013denominator relationships. Quantities in the numerator on both sides of the equal sign are directly proportional whereas quantities in the numerator on one side and the denominator on the other side are inversely proportional.<\/li>\r\n \t
    2. Notice how the variables are modified (e.g., by exponents, logarithms, numerical constants) and determine what that indicates about the relationships between the variables.<\/li>\r\n<\/ol>\r\n
      \r\n\r\nLet\u2019s apply this procedure to understanding Einstein\u2019s equation. Answer the following questions related to Equation\u00a0(32)<\/span><\/a>.\r\n\r\n<\/div>\r\n
        \r\n \t
      1. What does each variable (E, m, c) represent?<\/li>\r\n \t
      2. Which is variable and which constant?<\/li>\r\n \t
      3. What are the dimensions for energy?<\/li>\r\n \t
      4. What are the dimensions for mass?<\/li>\r\n \t
      5. How are energy and mass related to each other?<\/li>\r\n \t
      6. How are the variables modified? Are there any exponents?<\/li>\r\n<\/ol>\r\n
        \r\n\r\nWhen we put all of this together, we see that energy and mass are directly proportional because both are in the numerator, and that the constant of proportionality is the square of the speed of light. Since the square of the speed of light is such a large number, we can conclude that a small amount of mass<\/em> is equivalent to a large amount of energy<\/em>.\r\n\r\n<\/div>\r\n
        \r\n\r\nThe advantage of the equation over a sentence that says basically the same thing is that the equation allows us to directly calculate how much energy we can get from a given mass of matter. This gives us the two things that make equations important to us.\r\n\r\n<\/div>\r\n
          \r\n \t
        1. Equations express natural phenomena in a language that is widely understood and easy to communicate.<\/li>\r\n<\/ol>\r\n
            \r\n \t
          1. Equations allow us to calculate and make predictions about the magnitude of one parameter based on the values of other parameters.<\/li>\r\n<\/ol>\r\n
            \r\n\r\nNext, let\u2019s look at some equations that are a little closer to the subject matter of this book.\r\n\r\n<\/div>\r\n<\/div>","rendered":"
            \n
            \n

            For someone not used to dealing with equations and mathematical statements, the key to understanding them is to think about the physical and conceptual relationships that the equation expresses. Let\u2019s take a famous equation\u2014shown in Equation\u00a0(32)<\/span><\/a>\u2014and think about what it means conceptually.<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n
            <\/th>\n[latex]E = {mc}^{2}[\/latex]<\/th>\n(32)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n
            \n

            Where E represents energy, [latex]m[\/latex] represents mass, and [latex]c^{2}[\/latex] is a very large number: the square of the speed of light. What this equation means, in Einstein\u2019s words (Mayne, 1948), is:<\/p>\n<\/div>\n

            \n

            \u201cIt followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing\u2014a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c\u2011squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before.\u201d<\/p>\n<\/div>\n

            \n

            The point of the equation is that matter (i.e., mass) and energy are fundamentally the same thing. The equation tells us that energy is equal to mass times a very large number ([3.0\u00a0x\u00a0108<\/sup> m\/s]2<\/sup>), indicating that a small amount of mass is equal to a large amount of energy. This is a very significant concept, especially to the residents of Hiroshima and Nagasaki, Japan, during the mid\u20111940s (where 64 kg of enriched uranium was converted to 16 kilotons of energy).<\/p>\n<\/div>\n

            \n

            A useful approach to understanding equations and mathematics is to look at the relationships between the various parts of the equation and determine how those parts relate to reality. Here are basic steps for that approach.<\/p>\n<\/div>\n

              \n
            1. Look at each item in the equation and determine what it represents (e.g., length; density; velocity). It also helps to figure out the dimensions of each variable (e.g., length, L; density ML-3<\/sup>; velocity, LT-1<\/sup>) and their associated units (e.g., meters; kg\/m3<\/sup>; m\/s).<\/li>\n
            2. Determine if each item varies or is constant. Constants are items with a value that does not change such as gravitational acceleration on Earth, the speed of light in a vacuum, and the number of molecules in a mole of a substance.<\/li>\n<\/ol>\n
                \n
              1. Compare the variables on one side of the equation to those on the other side. Look for numerator\u2013denominator relationships. Quantities in the numerator on both sides of the equal sign are directly proportional whereas quantities in the numerator on one side and the denominator on the other side are inversely proportional.<\/li>\n
              2. Notice how the variables are modified (e.g., by exponents, logarithms, numerical constants) and determine what that indicates about the relationships between the variables.<\/li>\n<\/ol>\n
                \n

                Let\u2019s apply this procedure to understanding Einstein\u2019s equation. Answer the following questions related to Equation\u00a0(32)<\/span><\/a>.<\/p>\n<\/div>\n

                  \n
                1. What does each variable (E, m, c) represent?<\/li>\n
                2. Which is variable and which constant?<\/li>\n
                3. What are the dimensions for energy?<\/li>\n
                4. What are the dimensions for mass?<\/li>\n
                5. How are energy and mass related to each other?<\/li>\n
                6. How are the variables modified? Are there any exponents?<\/li>\n<\/ol>\n
                  \n

                  When we put all of this together, we see that energy and mass are directly proportional because both are in the numerator, and that the constant of proportionality is the square of the speed of light. Since the square of the speed of light is such a large number, we can conclude that a small amount of mass<\/em> is equivalent to a large amount of energy<\/em>.<\/p>\n<\/div>\n

                  \n

                  The advantage of the equation over a sentence that says basically the same thing is that the equation allows us to directly calculate how much energy we can get from a given mass of matter. This gives us the two things that make equations important to us.<\/p>\n<\/div>\n

                    \n
                  1. Equations express natural phenomena in a language that is widely understood and easy to communicate.<\/li>\n<\/ol>\n
                      \n
                    1. Equations allow us to calculate and make predictions about the magnitude of one parameter based on the values of other parameters.<\/li>\n<\/ol>\n
                      \n

                      Next, let\u2019s look at some equations that are a little closer to the subject matter of this book.<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":11,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1130","chapter","type-chapter","status-publish","hentry"],"part":1477,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":2,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1130\/revisions"}],"predecessor-version":[{"id":1716,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1130\/revisions\/1716"}],"part":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/parts\/1477"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1130\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/media?parent=1130"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapter-type?post=1130"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/contributor?post=1130"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/license?post=1130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}