Macaulay brackets

Macaulay brackets are a type of mathematical function that you can use to describe something called a "piecewise function". A piecewise function is just a fancy way of saying that you have different rules for calculating a value based on what the input value is.

Here's an ELI5 example: Imagine you're trying to figure out how much a sandwich costs based on how many ingredients are in it. If the sandwich has 1-3 ingredients, it costs $5. If it has 4-6 ingredients, it costs $7. And if it has 7 or more ingredients, it costs $10.

We could express this "rule" using a piecewise function, and that's where Macaulay brackets come in. Macaulay brackets are just a shorthand way of expressing those different "rules" for calculating the value of a piecewise function.

Here's the actual formula for a piecewise function using Macaulay brackets:

f(x) = {[value if condition is true], if [condition]; [value if next condition is true], if [next condition]; [value if final condition is true], if [final condition]}

Using our sandwich example, we could express the cost of the sandwich using Macaulay brackets like this:

f(x) = {5, if x <= 3; 7, if 3 < x <= 6; 10, if x > 6}

So if we put 2 ingredients in the sandwich, the input value (x) would be less than or equal to 3, and the output value (the cost of the sandwich) would be $5. If we put 5 ingredients in the sandwich, the input value would be between 3 and 6, and the output value would be $7. And if we put 7 ingredients in the sandwich, the input value would be greater than 6, and the output value would be $10.

Hope that helps!