Why Calculators Lie: Can You Solve This Simple Math Problem?

Dave's Garage
22 Aug 202311:47

TLDRIn 'Dave's Garage,' Dave Plummer, a retired software engineer, explores why calculators and people often get a simple math problem wrong. He uses the example of 6 divided by 2, multiplied by 2 plus 1 in parentheses, which incorrectly yields 1 on some calculators. Dave explains the correct order of operations (PEMDAS) and demonstrates how modern calculators and online tools correctly solve the problem to get 9, unlike older calculators that may not support parentheses. The video also touches on the history of calculators and the importance of following mathematical rules.

Takeaways

  • 🧮 The video discusses why calculators and people sometimes get a simple math problem wrong.
  • 👚‍💻 Dave Plummer, a retired software engineer from Microsoft, explains the issue with the calculator.
  • 🔢 The problem involves the order of operations in the expression '6 divided by 2(2+1)', which many calculators incorrectly evaluate as 1.
  • 📚 The correct approach is to follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) rules.
  • 🀔 The video highlights a common mistake where people keep parentheses after solving the expression inside them.
  • 🛠 Dave demonstrates the correct calculation, which should yield 9, not 1, by properly applying the order of operations.
  • 📱 Modern calculators, including those on Windows, Mac, and online platforms, correctly solve the problem when entered correctly.
  • 📚 The video also touches on historical calculators and how they handled expressions, with some requiring users to be explicit about the order of operations.
  • 💻 Dave explains the technical aspects of how computers process mathematical expressions, including infix and postfix notation.
  • 🔧 The video suggests that some older calculators and software like VisiCalc from 1979 required users to be explicit about the order of operations.
  • 📘 Dave references his book 'Secrets of the Autistic Millionaire', which is related to living a successful life with ASD, not about money.

Q & A

  • What is the main issue discussed in Dave's Garage video?

    -The main issue discussed is why many people and calculators get a simple math problem wrong, specifically the problem of '6 divided by 2(2+1)'.

  • Who is Dave Plummer and what is his background?

    -Dave Plummer is a retired software engineer from Microsoft, who has experience dating back to the MS-DOS and Windows 95 days, and he once owned the development of the Windows calculator.

  • What does the acronym PEMDAS stand for and what is its significance?

    -PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). It is significant as it helps remember the order of operations to properly solve mathematical expressions.

  • What is the correct answer to the math problem '6 divided by 2(2+1)' and why?

    -The correct answer is 9. According to the PEMDAS rule, you first solve the expression within the parentheses (2+1), which equals 3, then you perform the division (6/2), and finally, you multiply the result by the number in the parentheses (3*3).

  • Why does Dave's old Sharp EL-5103S calculator give the wrong answer for the problem?

    -The Sharp EL-5103S calculator gives the wrong answer because it does not correctly apply the PEMDAS rule by not removing the parentheses after calculating the expression inside them, leading to an incorrect order of operations.

  • What is infix notation and how does it relate to the discussed math problem?

    -Infix notation is a mathematical notation where the operator is placed between the operands, as in the standard algebraic expressions. The discussed math problem is an example of infix notation, which can be ambiguous and is why calculators and computers often convert it to postfix notation for easier processing.

  • How does Windows Calculator handle the math problem '6 divided by 2(2+1)' in the video?

    -Windows Calculator, starting from Windows 3.1, correctly handles the math problem by requiring the user to explicitly insert the multiplication operator, which ensures the correct order of operations and results in the correct answer of 9.

  • What is the difference between infix and postfix notation, and why is postfix easier for computers?

    -Infix notation places the operator between the operands, while postfix notation places the operator after the operands. Postfix notation is easier for computers because it can be represented as a stack, which computers can process more efficiently using a simple stack algorithm.

  • What is RPN (Reverse Polish Notation) and why is it popular among engineers?

    -RPN is a mathematical notation where the operator follows the operands, making it unambiguous and precise. It is popular among engineers because it eliminates the need for parentheses and allows for a clear, straightforward order of operations.

  • How does the video address the argument that the expression '6 divided by 2(2+1)' should equal 1?

    -The video explains that the correct interpretation follows the PEMDAS rule, and that the expression should be evaluated as '6 divided by 2 times 3', which equals 9. The argument for an answer of 1 is based on an outdated rule and is not applicable in modern mathematics.

Outlines

00:00

🔢 The Misunderstood Math Problem

Dave Plummer, a retired software engineer from Microsoft, begins the video by addressing a common math problem that often leads to confusion and debate. He discusses his background, including his involvement in the development of Windows calculator during the transition to infinite precision math libraries. Dave then introduces his long-time primary calculator, a Sharp EL-5103S, and demonstrates how it incorrectly calculates the expression '6 divided by 2 times 2 plus 1', yielding 1 instead of the correct answer, 9. He explains the importance of following the correct order of operations, known as PEMDAS, to arrive at the right solution and points out that modern calculators and online tools correctly solve the problem by adhering to these rules.

05:02

👚‍💻 Historical Calculator Accuracy and Notation

In this section, Dave explores the evolution of calculators and their accuracy in handling mathematical expressions. He tests various versions of Windows calculators, starting from Windows 1.01, which lacks support for parentheses, up to Windows 95, which correctly processes the expression using parentheses due to its input method. Dave delves into the complexities of mathematical expression parsing, explaining infix and postfix notations and how computers use stack algorithms to evaluate expressions. He also touches on the use of Reverse Polish Notation (RPN) in older calculators, particularly favored by engineers for its precision and unambiguity. Dave concludes by examining the capabilities of early software like VisiCalc and modern online calculators, which correctly interpret and solve the given expression by explicitly applying the multiplication operator.

10:02

📚 Correcting Math Misconceptions

Dave concludes the video by addressing the remaining misconceptions about the math problem and the correct way to solve it. He refutes the idea of an 'inline division' rule that was outdated by 1915, emphasizing the importance of following PEMDAS. Dave suggests an alternative method of solving the expression by rewriting it to clearly show the division by a factor of two, leading to the correct answer of 9. He light-heartedly requests viewers to rewatch the video if they still disagree and humorously references Gerald Undone. Dave ends by inviting viewers to subscribe to his channel for more content and promotes his book, 'Secrets of the Autistic Millionaire', which offers insights into living a successful life on the Autism Spectrum.

Mindmap

Keywords

💡Calculator

A calculator is an electronic device used to perform arithmetic operations and basic mathematical functions. In the context of the video, it's highlighted that calculators can sometimes provide incorrect answers due to the way they parse and calculate mathematical expressions, especially when following the order of operations.

💡PEMDAS

PEMDAS is an acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It is a mnemonic used to remember the order of operations in mathematics. The video emphasizes the importance of following PEMDAS to correctly solve the presented math problem.

💡Infinite Precision

Infinite precision refers to the ability of a calculator or computer system to handle numbers with an unlimited number of digits without rounding errors. The video mentions the transition to infinite precision in the development of Windows calculator, which was a significant change in how calculations were performed.

💡Pentium Floating Point Bug

The Pentium Floating Point Bug refers to a flaw in the floating-point unit of Intel's Pentium processors that caused incorrect floating-point arithmetic results under certain conditions. The video script mentions it as a potential disaster similar to the incorrect calculations in calculators.

💡Sharp EL-5103S

The Sharp EL-5103S is a model of calculator that the video creator has been using for over 40 years. It is highlighted as a reliable device that, despite its age, still functions well and is the creator's primary calculator, although it has been shown to incorrectly solve certain expressions.

💡Parentheses

Parentheses are symbols used in mathematical expressions to indicate the order in which operations should be performed. In the video, the correct handling of parentheses is crucial to solving the math problem correctly, and the misuse of parentheses leads to the incorrect answer provided by some calculators.

💡Infix Notation

Infix notation is the conventional mathematical notation where operators are placed between their operands. The video discusses how infix notation can be confusing for calculators to process and how it is converted into postfix notation for easier computation by computers.

💡Postfix Notation

Postfix notation, also known as Reverse Polish Notation (RPN), is a way of laying out operators and operands where operators follow their operands. The video explains that postfix notation is easier for computers to process using a stack algorithm, which simplifies the evaluation of expressions.

💡Expression Evaluation

Expression evaluation refers to the process of calculating the value of a mathematical expression. The video demonstrates the correct way to evaluate the given expression by following the PEMDAS rule and contrasts it with the incorrect method that leads to the wrong answer.

💡Microsoft

Microsoft is the company where the video creator, Dave Plummer, worked as a software engineer. He mentions his role in the development of the Windows calculator during the transition to infinite precision, which is relevant to the discussion of calculator accuracy.

💡Online Calculators

Online calculators are web-based tools that perform mathematical calculations. The video compares the performance of various online calculators, such as Google's, in evaluating the given expression correctly by adhering to the proper order of operations.

Highlights

Dave Plummer, a retired software engineer from Microsoft, discusses common calculator errors in simple math problems.

Introduction to the historical development of Windows Calculator and its transition to infinite precision math libraries.

The presenter's long-term use of a Sharp EL-5103S calculator since 1982 and its reliability.

Demonstration of a viral math problem that often yields incorrect results on calculators: 6÷2(2+1).

Explanation of the correct order of operations using PEMDAS and why many calculators get the problem wrong.

Dave's exploration of calculators from Windows 1.01 to Windows 95 and their handling of the viral math problem.

The importance of discarding parentheses and converting them to multiplication after solving expressions within them.

How infix notation is converted to postfix for easier computer processing using a stack algorithm.

The presenter's comparison of infix notation with postfix and the advantages of postfix for computers.

A look back at the use of Reverse Polish Notation (RPN) in older calculators and its popularity among engineers.

Testing the VisiCalc software from 1979 and its correct handling of the viral math problem.

Modern online calculators like Google's correctly solving the problem by explicitly inserting the multiplication step.

Explanation of how Chat GPT 4 evaluates the viral math problem step by step using PEMDAS.

Historical perspective on an old mathematical rule that is no longer applicable to the viral math problem.

Dave's suggestion to consider the expression as 6/2 times the parenthetical, leading to the correct answer of 9.

A humorous request for viewers to watch the video again at 75% speed for better understanding.

Invitation to subscribe to Dave's channel for more informative content and a mention of his book on living successfully with Asperger's.

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