Understanding and Solving Complex Mathematical Expressions: The Case of 17 - 17 * 17 / 17

When faced with complex mathematical expressions, understanding the order of operations is crucial. This article explores how to solve the expression 17 - 17 * 17 / 17 using the correct order of operations, known as PEMDAS or BODMAS. We will also delve into alternative evaluation methods and why they might be considered.

Solving 17 - 17 * 17 / 17 Using PEMDAS/BODMAS

Let's break down the given expression using the popular mnemonic PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

The expression in question is 17 - 17 * 17 / 17.

Multiplication and Division First

17 * 17 289

17 / 17 1

Substituting these values back into the expression:

17 - 289 1

Performing Addition and Subtraction from Left to Right

17 - 289 -272

-272 1 -271

Therefore, the final result is -271.

Alternative Evaluation Methods: Left to Right Order

Another method of evaluating this expression involves working from left to right without following the standard order of operations. This method, however, is less commonly used and may not yield the correct result in most cases.

17 - (17 * 17 / 17)

First, perform the multiplication and division:

17 - (289 / 17)

289 / 17 17

Substitute this back into the expression:

17 - 17 0

So, according to this method, the result is 0.

A Deeper Look at Algebraic Systems

The concept of alternative algebraic systems is discussed by Tim Gowers in his book Mathematics: A Very Short Introduction. While the standard algebraic system, following PEMDAS/BODMAS, is widely accepted, it is important to understand that other systems can theoretically exist as long as they maintain logical consistency.

For instance, consider the expression 4 3 * 2. In the standard system, this evaluates to 10. However, some might argue for evaluating addition before multiplication, leading to:

4 (3 * 2) 4 6 10

But in another algebraic system, you could evaluate the expression as:

(4 3) * 2 7 * 2 14

While this system may not be practical, it demonstrates the versatility of mathematical systems.

Conclusion

Understanding the correct order of operations is essential for accurate mathematical evaluation. While alternative methods can exist, it is important to follow the widely accepted standards like PEMDAS or BODMAS for consistency and correctness. Exploring these concepts further can provide a deeper appreciation for the flexibility and depth of mathematical systems.