Cracking the Code: Uncommon Concepts in Mathematics

Mathematics, a field of study cherished for its clarity and precision, often hides a multitude of concepts not yet fully appreciated by the broader public. Among these, the Order of Operations, the concept of a Function, and the intriguing topic of Randomness stand out for their depth and importance. Let's explore these key mathematical ideas in more detail.

The Order of Operations

The Order of Operations, commonly referred to as PEMDAS (or BODMAS in some regions), is a fundamental principle in mathematics. However, despite its frequent introduction in early education, it often remains a mystery beyond the basics. The Order of Operations dictates the sequence in which operations should be performed to solve mathematical expressions accurately.

Understanding the Sequence

Order of Operations (PEMDAS/BODMAS): Parentheses/brackets, Exponents/roots, Multiplication/Division, Addition/Subtraction. Solving an Expression: Start by solving what’s inside the parentheses or brackets. Each set gets its own instance of PEMDAS/BODMAS. Next, evaluate exponents or roots. Multiplication and division are then performed from left to right, as are addition and subtraction for the same reason.

For example, let's solve the equation: 53 - 32 ÷ log10(27) × 12 - 4 x.

Solve the parentheses or brackets, if any. Here, the parentheses need to be addressed, followed by working through the expression inside. In this case, 32 and log10(27), notably, log10(27) 3 (since 103 27). The equation now becomes: 53 - 9 ÷ 3 × 12 - 4 x. Next, evaluate multiplication and division from left to right. 9 ÷ 3 3, and 3 × 12 36. The equation now is: 53 - 36 - 4 x. Perform addition and subtraction left to right. 53 125, and 125 - 36 - 4 85. Thus, x 85.

The Function

A mathematical function is a concept with deep philosophical underpinnings. It is formally defined as a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. Functions are often visualized on a coordinate plane (the Cartesian coordinate system) where the x-axis (horizontal) represents the input and the y-axis (vertical) represents the output.

A true function cannot have any vertical sections where the line is infinite. Here is an interesting real-world example using the US Government's Social Security Number program. If you input a Social Security Number into a database, you will get only one unique name as an output.

Randomness and Its Complexity

Randomness is a subject far more complex than it appears. The mathematical treatment of randomness can explore deep areas such as probability theory, statistics, and information theory. For a thorough understanding, one would need to delve into the nuances of these topics, much like in other areas of mathematics.

Additional Insights

Relative vs Absolute

The concepts of relative versus absolute are crucial in decision-making processes. For instance, consider the example of choosing between two sizes of potato chips. Bag A (5 oz for $1.29) costs 25.8 cents per oz, while Bag B (13 oz for $2.99) costs 23 cents per oz. From an absolute standpoint, Bag B is more cost-effective unit-wise. However, from a relative standpoint, if you have enough money to buy Bag B, it could be more practical to get the larger size for better value.

Temperature Scales and Units

Temperature scales also pose interesting challenges. Fahrenheit and Celsius are relative scales, while Rankine and Kelvin are absolute. The "zero" points for Fahrenheit and Celsius are arbitrarily chosen. Absolute zero, however, is the same in both Rankine and Kelvin. When dealing with scientific formulas involving temperatures, it's critical to use Rankine or Kelvin to avoid errors.

Accuracy vs Precision

Accuracy versus precision is another critical aspect of measurement and data interpretation. Precision refers to the degree of agreement among multiple measurements, while accuracy pertains to how close a measurement is to the true or accepted value. The example provided illustrates this well. If a measuring device is accurate only to the tenth place, a measurement of 0.05 would be rounded up to 0.1.

Understanding these concepts provides a deeper insight into the intricacies of mathematics and can help in making more informed decisions and solving complex problems.