🔫Added algorithm explanation for ceil

en/Basic Math/Ceil.md

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+# Ceil
+
+The ceiling function is a mathematical function which rounds a given real number $x \in \mathbb{R}$ up to the nearest integer. It is commonly referred to as the `ceil` function. We can also restate the function to say that `ceil(x)` gives us the smallest integer greater than $x$.
+
+So, if $x \in \mathbb{Z}$ is an integer, then `ceil(x)` returns $x$. However, if the number has any decimnal component, it will round up to the next integer, regardless of the magnitude of the decimal part.  
+
+For positive numbers $x \in \mathbb{R}^{+}$, `ceil(x)` rounds up to the next integer. So, for example, `ceil(6.9)` returns `7`.  
+
+For negative numbers $x \in \mathbb{R}^{-}$, `ceil(x)` rounds up towards zero, just like for positive numbers. This does mean that the absolute value of `ceil(x)` is less than that of $x$. So, for example, `ceil(-6.9)` returns `-6`, as $-6 > -6.9$.  
+
+## Idea and Implementation
+
+For **languages which have separate integer and floating types** (C, C++, Java, Python, Rust, Zig), the `ceil` function takes in a floating point number `x` and returns the integer, which is the result of `ceil(x)`.
+
+When the floating point number `x` is typecasted into an integer, the digits after the decimal points are cut off without rounding. So, on typecasting `x` into an integer, we get just the integer part and the fractional part is disregarded. So, `int(6.9)` gives us `6` and `int(-6.9)` gives us`-6`. In the case where `x` is negative or zero, that actually is enough for us, as that is the value of `ceil(x)` as well! And as for when `x` is positive, `int(x)` is actually 1 less than `ceil(x)`, and we can just define add one to `int(x)` to get the value of `ceil(x)`. So the pseudocode for the same is as stated below.
+
+```py
+function ceil(x: float)
+    if x > 0
+        return int(x) + 1
+    else
+        return int(x)
+end
+```
+
+In the meanwhile, for **languages which just have one single type for real numbers** (JavaScript), the `ceil` function takes in a number `x` and needs to return a number which is the output of the mathematical function `ceil(x)`. This can be implemented by either using the truncation functions provided natively by the language, which would follow the previous implementation by replacing typecasting with truncation.
+
+Otherwise, another implementation which can be taken is by getting the decimal part by using x (modulo 1), i.e., `x % 1`, which will return the digits after the decimal point for `x` (when `x` is positive or zero), which can be expressed as `x - (ceil(x) - 1)` and will return `x - ceil(x)` (when `x` is negative). The former expression can be understood by the fact that `ceil(x) - 1` returns the integral part of `x` for positive values of `x` and when subtracting that from `x`, we get the digits after the decimal point of `x`, which is what `x % 1` returns. The latter expression can be understood by the fact that for negative values of `x`, `ceil(x)` returns the integral part of `x` after truncating of the digits after decimal point. Hence, we get the digits after the decimal point after subtracting `ceil(x)` from `x`.
+
+So, now we can evaluate `ceil(x)` to be equal to `x + 1 - (x % 1)` for positive and zero values of `x` and `x - (x % 1)` for negative values of `x`.
+
+```js
+function ceil(x)
+    if x >= 0
+        return x + 1 - (x % 1)
+    else
+        return x - (x % 1)
+end
+```
+## Walkthrough
+
+Let us walk through how the algorithm works for positive and negative numbers by using the numbers $x_1 = 6.9$ and $x_2 = -4.2$.
+
+For the first implementation:
+
+* $x_1$ is positive, and we get $int(x_1)$ to be $6$. And going by the algorithm, we are returned $6 + 1 = 7$, which is the correct output.
+* $x_2$ is negative, and we get $int(x_2)$ to be $-4$. And going by the algorithm, we are returned $-4$, which is the correct output.
+
+For the second implementation:
+
+* $x_1$ is positive. So `x % 1` gives us `0.9`. Thus `x + 1 - (x % 1)` gives us `6.9 + 1 - 0.9` which is `7`, which is the correct output.
+* $x_2$ is negative. So `x % 1` gives us `-0.2`. Thus `x - (x % 1)` evaluates to `-4.2 - (-0.2)`, which is `-4`, which is the correct output.
+
+## Applications
+
+* Quadratic reciprocity
+* Rounding of numbers
+
+## Resources
+
+* [Wikipedia Article](https://www.wikiwand.com/en/Ceil)