1.4.5. Controlling the hierarchy of operations or precedence

Let’s consider the previous arithmetic operation, but now we will include parentheses. For example, 1+2 × 3 will become (1+2) × 3

(1+2)*3 ans = 9

In MATLAB, the order of operations can be controlled using parentheses. When you include parentheses around certain parts of an expression, you are telling MATLAB to perform the operations inside the parentheses first. In this example, (1 + 2) is calculated to be 3, and then that result is multiplied by 3, leading to a final answer of 9.

Think of using parentheses like organizing tasks in a to-do list. If you prioritize certain tasks (put them in parentheses) and do them first, you'll get things done in a specific order. For instance, if you say, 'First buy groceries (which includes getting a list) and then cook dinner', doing the grocery task first ensures you have everything ready for dinner.

The order in which MATLAB performs arithmetic operations is exactly that taught in high school algebra courses. Exponentiations are done first, followed by multiplications and divisions, and finally by additions and subtractions. However, the standard order of precedence of arithmetic operations can be changed by inserting parentheses. For example, the result of 1 + 2 × 3 is quite different than the similar expression with parentheses (1 + 2) × 3.

MATLAB follows a specific order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means it will first solve anything in parentheses, next any exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. When you use parentheses, you override this natural precedence, which can lead to different results.

Imagine baking a cake. If you read the recipe carefully, some steps must happen before others—like mixing dry ingredients before adding wet ones. The order matters in cooking just as it does in math; doing things out of order can lead to mistakes. If you add eggs (addition) before your dry ingredients (multiplication) incorrectly, your cake might not turn out right.

To make the evaluation of expressions unambiguous, MATLAB has established a series of rules. The order in which the arithmetic operations are evaluated is given in Table 1.2.

To avoid confusion over how expressions are calculated, MATLAB has rules that define the precedence of operations. These rules ensure that everyone using MATLAB gets the same results from the same expressions. It's crucial to understand these rules to communicate ideas effectively and to avoid mistakes in calculations.

Consider a set of traffic signals controlling an intersection. If drivers adhere to a fixed set of rules about which signals to follow first, the flow of traffic is smooth. If everyone interpreted the signals in their own way, it could lead to chaos and accidents. Similarly, adhering to the order of operations keeps calculations predictable and reliable.

Therefore, we want to emphasize the importance of precedence rule in order to avoid ambiguity. For example: 1 / (2 + 3^2) + 4 / 5 * 6 / 7 gives ans = 0.7766 and without parentheses, 1 / 2 + 3^2 + 4 / 5 * 6 / 7 gives ans = 10.1857.

In this example, using parentheses changes the result significantly. The first expression calculates (2 + 3^2) first, which results in 11, then takes 1 divided by that result. In contrast, the second expression calculates 3^2 first and then processes the entire equation without considering which operations to prioritize, leading to a higher final result. This highlights that the absence of parentheses can lead to quite different answers.

Imagine planning a party. If you say, 'Invite Sarah after the guests arrive from the airport,' you are following a specific order that ensures everything goes smoothly. If you only say, 'Invite Sarah and guests,' it may be unclear when each should happen, leading to confusion. Just like that, unclear mathematical expressions can result in unexpected answers.

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