MATH 046 Lesson 2: Calculators

Entering Arithmetic Expressions into Calculators

• General Principle of Expression Entry:

  • Arithmetic expressions are entered into a calculator in the exact same syntax and order as they are written on paper.

  • Standard order of operations (PEMDAS / BDMAS) is applied automatically by the calculator when evaluating continuous expressions.

• Parentheses in Fractional Expressions:

  • When using the division sign ($/$ or $\div$) to represent fractions, parentheses must be placed around the entire numerator or the entire denominator whenever either contains an operation (such as addition or subtraction).

  • Omitting parentheses causes the calculator to perform division prior to addition or subtraction, producing an incorrect value for the fraction.

• Comparison of Unparenthesized vs. Parenthesized Input:

  • Unparenthesized Entry:

  • Expression: $5 + 3 \div 2$ (or $5 + 3 / 2$)

  • Calculation order: The calculator performs division first ($3 \div 2 = 1.5$) and then adds $5$.

  • Final output: $6.5$

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  • Parenthesized Entry:

  • Expression: $(5 + 3) \div 2$ (or $(5 + 3) / 2$)

  • Calculation order: The calculator evaluates the numerator operation first ($5 + 3 = 8$) and then divides by $2$.

  • Final output: $4$

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Key Input Errors and Solutions

• **Distinguishing Between Subtraction -$ $ and Negative (-) Keys**:\n - Calculators possess two distinct keys: one for binary subtraction (-$) and one for unary negation ($(-)).\n - Entering the subtraction symbol in place of a negative sign causes the calculator to throw a syntax error.\n - **Example Comparison**:\n - Correct entry using negative sign key: 8 + (-3) = 5\n - Incorrect entry using subtraction key: 8 + -3 \rightarrow \text{Error}\n - \n\n- **Order of Operations with Exponents and Negation**:\n - Evaluated order of operations inside a calculator processes parentheses first, exponents second, and unary negation third.\n - **Unparenthesized Base**: \n - Expression: -3^2\n - Interpretation: The calculator evaluates -(3^2)$. It calculates$3^2 = 9 first, then negates the result.\n - Result: -9\n - **Parenthesized Base**: \n - Expression: (-3)^2\n - Interpretation: The calculator treats the full term -3$as the base being squared ($-3 \times -3).\n - Result: 9\n\n# Step-by-Step Problem Calculations\n\n- **Evaluating Practice Expressions**:\n - **Problem 1**: Compute (-3)^2 - 5^2\n - Step 1: Evaluate (-3)^2 = 9\n - Step 2: Evaluate 5^2 = 25\n - Step 3: Compute 9 - 25 = -16\n - **Problem 2**: Compute -15 \div 5(-3)\n - Step 1: Perform left-to-right division: -15 \div 5 = -3\n - Step 2: Perform multiplication: -3 \times (-3) = 9\n - **Problem 3**: Compute 5^2 - (3^2 + 4^2)\n - Step 1: Evaluate terms inside parentheses: 3^2 = 9$and$4^2 = 16\n - Step 2: Add parenthesized terms: 9 + 16 = 25\n - Step 3: Evaluate leading square: 5^2 = 25\n - Step 4: Subtract: 25 - 25 = 0\n - **Problem 4**: Compute \frac{(4 \times 5)(1 \times 12)}{(5 \times 2 \times 3)}\n - Step 1: Compute numerator product: (4 \times 5) \times (1 \times 12) = 20 \times 12 = 240\n - Step 2: Compute denominator product: 5 \times 2 \times 3 = 30\n - Step 3: Perform division: \frac{240}{30} = 8\n\n# Sequential Calculation Puzzle and Calculator Word Game\n\n- **Sequential Step-by-Step Problem**:\n - **Step 1**: Calculate 28 \div 4 - (4 + 3)\n - Parenthesis evaluation: 4 + 3 = 7\n - Division evaluation: 28 \div 4 = 7\n - Subtraction: 7 - 7 = 0\n - **Result 1**: 0\n - **Step 2**: Calculate 22 + 16 - 8\n - Addition: 22 + 16 = 38\n - Subtraction: 38 - 8 = 30\n - **Result 2**: 30\n - **Step 3**: Add the answers from Step 1 and Step 2\n - Computation: 0 + 30 = 30\n - **Result 3**: 30\n - **Step 4**: Add 5,305 to Result 3\n - Computation: 30 + 5,305 = 5,335\n - **Final Numerical Output**: 5,335\n\n- **Calculator Word Riddle Solution**:\n - **Riddle**: "Sometimes an alligator \_\_\_\_\_ in wait for its prey."\n - **Inversion Decodification**:\n - When turning a seven-segment calculator display showing 5335 upside down:\n - Digit 5 reads as `S`\n - Digit 3 reads as `E`\n - Digit 3 reads as `E`\n - Digit 5$reads as `L` (reading inverted string$5335$$\rightarrow LIES)

  • Decoded Word: LIES

  • Completed Statement: "Sometimes an alligator LIES in wait for its prey."