Before presenting the fundamental laws it should be noted there are four basic mathematic operations with numbers and these are: addition, subtraction, multiplication, and division. The afformentioned fundamental laws apply to these four mathematical operations.
The fundamental laws in mathematics are:
- Commutative Law for Addition - \(a+b = b+a\),
- Associative Law for Addition - \((a+b)+c = a + (b+c)\),
- Identity Law for Addition - \(a+0=0+a\),
- Inverse law for Addition - \(a + (-a) = (-a)+a = 0\),
- Associative Law for Multiplication - \(a(bc) = (ab)c\),
- Inverse Law for Multiplication - \(a\left(\frac{1}{a}\right) = \left(\frac{1}{a}\right)a, \quad a\neq 0\),
- Identity Law for Multiplication - \((a)(1) = (1)(a) = 0\),
- Commutative Law for Multiplication - \(ab = ba\), and
- Distributive Law - \(a(b+c) = ab + ac\)
The order of arithmetic operations are:
- simplify terms inside parentheses or brackets,
- simplify the exponents and roots,
- perform multiplication and division,
- perform addition nad subtraction.
Examples - Fundamental Properties
- Example 1 - Calculate the expression $$ 71 - \{-86+97-\left[84+37-99+\left(-3-15+29\right)\right]-71\}$$
Solution $$\begin{eqnarray}&71& - \{-86+97-\left[84+37-99+\left(-3-15+29\right)\right]-71\} = \\\nonumber &=& 71 - \{11-\left[22+11\right]-71\} = \\ \nonumber &=& 71 -\{11-33-71\} =\\\nonumber &=& 71 + 22 + 71 = \\ \nonumber &=& 164 \end{eqnarray}$$ - Example 2 - Calculate the expression $$ -31-\{78-56+\left[-32-5-(7-8+2)-3\right]+4-5\}$$
Solution $$ \begin{eqnarray}&-31&-\{78-56+\left[-32-5-(7-8+2)-3\right]+4-5\} = \\ \nonumber &=& -31 - \{22+\left[-37 - 1-3\right] - 1\} = \\ \nonumber &=& -31 -{21-41} = \\\nonumber &=& -31+20 = -11\end{eqnarray}$$ - Example 3 - Calculate the expression $$ \frac{5}{3}-\frac{13}{8}:1.5 + \frac{5}{12}$$ Solution $$ \begin{eqnarray}&\frac{5}{3}&-\frac{13}{8}:\frac{3}{2} + \frac{5}{12} = \\ \nonumber &=& \frac{5}{3}-\frac{13}{8}\cdot \frac{2}{3} + \frac{5}{12} = \\ \nonumber &=& \frac{5}{3}-\frac{13}{12} + \frac{5}{12} = \\ \nonumber &=& \frac{5}{3} - \frac{8}{12} = \\\nonumber &=&\frac{5}{3} - \frac{2}{3} = \frac{3}{3} = 1\end{eqnarray}$$
- Example 4 - Determine the three rational numbers between \(\frac{1}{9}\) and \(\frac{1}{7}\).
Solution - The boundaries in decimal form have to be determined first i.e. \(\frac{1}{9} = 0.1111\) and \(\frac{1}{7} = 0.142857.\) Generally there could be any number between those two, however, here are some examples: $$ 0.12, 0.13, 0.14 $$ - Example 6 - Let's say that \(n\geq 9\) and that \(n\) is the natural number. Determine which one of the following numbers is the biggest.
- \(a = \frac{9}{n}\)
- \(b = \frac{n}{9}\)
- \(c = \frac{n+1}{9}\)
- \(d = \frac{9}{n-1}\)
In case when \(n > 9\) lets say that \(n=10\) the following solutions are obtained. \begin{eqnarray} a &=& \frac{9}{10} = 0.9, \\ b &=& \frac{10}{9} = 1.111, \\ c &=& \frac{n+1}{9} = \frac{11}{9} = 1.222,\\ d &=& \frac{9}{n-1} = \frac{9}{9} = 1.\end{eqnarray} The biggest number was obtained in third case i.e. \(c = 1.222\) in case where \(n=10.\) - Example 7 - Calculate the expression \(0.5 - \frac{7}{2}.\)
Solution $$0.5-\frac{7}{2} = \frac{1}{2}-\frac{7}{2} = -\frac{6}{2} = -3.$$ - Example 8 - Calculate the expression \(\frac{0.05}{0.1}.\)
Solution Write 0.05 and 0.1 as fractions i.e. \(0.05 = \frac{5}{100},\) and \(0.1 = \frac{1}{10}.\) $$\frac{0.05}{0.1} = \frac{\frac{5}{100}}{\frac{1}{10}} = \frac{5}{10} = \frac{1}{2} = 0.5.$$ - Example 8 - Calculate the expression \(\frac{-7+5\cdot 9}{7:2-1}.\)
Solution - In this case first mathematical operations that need to be solved are multiplication and division. After these operations are the operations of addition and subtraction. Follwoing the order of mathematical operations the following solution is obtained $$ \frac{-7+45}{\frac{7-2}{2}} = \frac{38}{\frac{5}{2}} = \frac{76}{5} = 15.2$$ - Example 9 -Which of the offered values has a fraction \(\frac{231}{630}\) ?
- \(\frac{11}{90}\)
- \(\frac{7}{30}\)
- \(\frac{11}{30}\)
- \(\frac{7}{10}\)
Solution - The common divisor for both values is 21 the fraction is then simplified as follows: $$ \frac{231}{630} = \frac{\frac{231}{21}}{\frac{630}{21}} = \frac{11}{30}$$ The solution is \(c = \frac{11}{30}.\) - Example 10 - Which one of the following numbers belongs to the set of irrational numbers ?
- 4.33
- \(-\sqrt{16}\)
- \(-\frac{4}{7}\)
- \(\sqrt{5}\)
Solution - By definition an irrational number is a number which cannot be expressed as the ratio of two integers. The first given solution \(4.33\) is the number with decimal point but it can be expressed as the fraction \(\frac{433}{100}.\) The second solution \(-\sqrt{16}\) is equal to -4 so it is not irational rahter real number. The third given solution \(-\frac{4}{7}\) is the rational number. The final fourth number \(\sqrt{5}\) is equal to \(2.236067977499789696...\) and this is the irrational number since it cannot be expressed as the ratio of two integers. - Example 11 - Calculate the expression \(\frac{1+3\cdot(1.5-1)}{0.1 - 2\frac{3}{5}}\).
Solution the first step in the order of arithmetic operations is to simplify the terms inside the brackets and parenthesis. By doing so the following expression is obtained $$ \frac{1+3\cdot(1.5-1)}{0.1 - 2\frac{3}{5}} = \frac{1+3\cdot \left(\frac{3}{2} - 1\right)}{\frac{1}{10} - \frac{13}{5}} = \frac{1+3\cdot \frac{1}{2}}{\frac{1}{10} - \frac{13}{5}}$$ Then the next step is to perform multiplications and divisions and then addition and subtraction. $$ \frac{1+3\cdot \frac{1}{2}}{\frac{1}{10} - \frac{13}{5}} = \frac{1+1.5}{\frac{1-26}{10}} = \frac{\frac{25}{10}}{-\frac{25}{10}} = -1$$ - Example 12 - Calculate the expression \(\frac{0.001^2}{100\cdot 0.1}\)
Solution $$\frac{0.001^2}{100\cdot 0.1} = \frac{10^{-6}}{10^2\cdot 10^{-1}} = \frac{10^{-6}}{10^1} = 10^{-6-1} = 10^{-7}.$$ - Example 13 - Calculate the expression \(\frac{1 + 4.5\cdot \frac{1}{3}}{\left(2:0.1-4\right)\cdot 0.125}\)?
Solution $$\frac{1 + 4.5\cdot \frac{1}{3}}{\left(2:0.1-4\right)\cdot 0.125} = \frac{\frac{5}{2}}{\left(20-4\right)\cdot 0.125} = \frac{\frac{5}{2}}{16\cdot \frac{1}{8}} = \frac{\frac{5}{2}}{2} = \frac{5}{4} = 1.25$$
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