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Top 10 Mathematic Ideas Every Programmer
Should Master
The Ultimate Guide to Math for Programmers: 10 Concepts You Need to Know
CodeCircuit · Follow
Published in JavaScript in Plain English
14 min read · Apr 23
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As the world becomes more technologically advanced, programming has become a
crucial skill that many people want to learn.
However, there is a common misconception that programming doesn’t require any
math knowledge. This statement is not entirely true. Although you can program
without math, learning math makes programming easier to understand, and it unlocks
the secrets of the entire physical universe.
However, magic isn’t real math. Developers often avoid learning math because it looks
scary, but they fail to realize that math is the foundation of all programming, from
simple code to complicated programs like computer graphics and neural networks.
Understanding math concepts is what separates a true engineer from a novice.
We are no math genius. We are the opposite, which makes us uniquely qualified to
teach the ten essential math concepts that any programmer needs to understand.
1. Numeral Systems
Photo by Mika Baumeister on Unsplash
Throughout history, humans have used various methods to count things, but most of
them have been based on the number ten. We have ten fingers on our hands, and this
is what we use to count. For example, in the number 423, the four represents four
times one hundred in the hundreds place, the two represents two times ten in the tens
place, and the three represents three times one in the one’s place.
However, computers work off of a two-numerical system, also known as binary. In
base 2, the concept of place value works the same way as base 10, but because there
are only two symbols to work with, each place is multiplied by two. The first few
numbers in base 2 are 1, 2, 4, 8, 16, 32, 64, 128, and so on. Each number is double the
previous one.
Understanding base two is crucial for understanding other numeral systems like
hexadecimal, which is base 16. Hexadecimal uses the digits 0 through 9 along with A
through F to represent numbers. For example, the number A in hexadecimal
represents the number ten in base 10. Hexadecimal is commonly used to describe
binary values more concisely because each hexadecimal digit can translate into four
bits.
The base64 numeral system presents a more elaborate encoding scheme by
introducing additional letters and symbols for binary values. Consequently, developers
can convey binary data, including images, as text strings. Its applications extend to
email attachments and web content encoding.
Computers employ base 2 yet still necessitate the representation of numbers in base 10
for human comprehension. Hence, numerical miniatures in applications and
interfaces feature base 10. The conversion
of numbers to base 2 becomes necessary to
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display them in the binary system.
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2. Boolean Algebra
Boolean algebra represents a fundamental construct in computer science and
programming, which is almost always utilized while writing code. It is a variant of
algebra that concerns itself with binary variables that can have only two possible
values — true or false.
In this discourse, we will deliberate on the rudimentary elements of Boolean algebra,
and how they are utilized in coding.
There exist three operators in Boolean algebra: conjunction, disjunction, and
negation.
The conjunction operator (AND) is employed to fuse two or more Boolean values,
and the resulting value is only true if all the input values are true.
On the flip side, the disjunction operator (OR) is applied to merge two or more
Boolean values, and the resulting value is true if any of the input values are true.
The negation operator (NOT) is utilized to reverse a Boolean value, and the
resulting value is true if the input value is false, and false if the input value is true.
To shed light on the concept of Boolean algebra let us assume a relatable example.
Envision that you are endeavoring to secure a significant other. We can utilize the
Boolean variables “affluent” and “attractive” to define you. If you are both wealthy and
good-looking, you will most likely procure a significant other.
Nevertheless, if you are neither affluent nor attractive, you may have to remunerate a
model significant other. If you are either affluent or attractive, you can still get a
significant other, but your selection will be more restricted. We can demonstrate this
logic in code using if statements, or we can portray it visually using Venn diagrams.
Photo by Tolga Ahmetler on Unsplash
A Venn diagram is a graphical representation of a group of elements. In this scenario,
we can use Venn diagrams to display the various combinations of Boolean values.
Truth tables are another method to visually represent Boolean values. A truth table is a
table that demonstrates all the conceivable combinations of input values and the
resulting output values.
3. Floating Points Numbers in Computer Science
When engaging with numerical values in the domain of computer science, one of the
most ubiquitous methods is utilizing floating-point numbers. These numerical entities
provide programmers with the option to balance precision and range, as the storage
capacity for these values is limited.
If one is unacquainted with floating-point numbers, one might question how a
computer could generate nonsensical results, such as adding 0.1 and 0.2 and arriving
at 0.30004. The explanation lies in the realm of science, where numbers can range
greatly depending on the scope of the measurement.
To avoid listing these numerical values in full, the scientific community uses scientific
notation, whereby an exponent elevates the number to a rounded value, creating a
more concise form of the numerical value. Computers operate similarly to floatingpoint numbers, given the constraints of the storage capacity.
In particular, single-precision floating-point numbers require 32 bits, whereas doubleprecision floating-point numbers, which are the default in programming languages
like Python and JavaScript, require 64 bits. This is why they are deemed “floatingpoint” numbers, as the numerical value does not have a fixed number of digits
preceding or succeeding the decimal point.
Floating point numbers are a widely used numerical format in programming.
However, specific values, such as 0.01, cannot be expressed precisely in this format
due to their binary nature. As a result, there can be miniature inaccuracies, commonly
referred to as rounding errors, when performing operations with floating point
numbers.
To avoid these inaccuracies, programmers must understand the limitations of floating
point numbers and create adjustments to their code accordingly. This may entail
altering the sequence of operations, employing a data type with a higher degree of
precision, or applying rounding functions to rectify any discrepancies. It is imperative
to acknowledge and address these issues to produce accurate and reliable results in
computational tasks.
4. Logarithms
Logarithmic functions are a fundamental concept in algebra, frequently taught in high
school mathematics courses. To better comprehend these functions, consider an
actual log, such as a long piece of wood. The log begins at a length of 16 feet and is
gradually sawn in half until it reaches a length of 2 feet. Observe how the line drawn
through the log curves gradually rather than being perfectly straight.
In computer science, this gradual curve is representative of the operations of several
algorithms, including binary search. However, the intriguing aspect is that if the
number of times the log was cut is known, raising 2 to the power of that value yields
the original length of the log. In this scenario, four cuts were used to achieve a length
of 2 feet; thus, raising 2 to the power of 4 provides the original length of the log. This is
known as exponentiation.
Conversely, the logarithm is the precise reverse of exponentiation. Assume that we are
unaware of the exponent but know the original length of the log, and we want to
determine the number of times the log must be cut to achieve a length of 2 feet. This
can be calculated using a base 2 log on the original length, which yields a value of 4.
When the base is 2, it is directed to a binary logarithm. However, in mathematics, the
common logarithm, which uses base 10, is more prevalent.
Logarithmic functions have applications beyond mathematics, including science,
engineering, and computer science. They assist in solving problems related to growth
rates, sound levels, and earthquake magnitudes, among others. Furthermore, they are
utilized in finance, such as calculating interest rates and compound growth.
5. Set Theory
Set theory may seem unfamiliar to those new to the field of computer science or
programming. However, in essence, it is the study of collections of distinct, unordered
entities. This article seeks to delve deeper into the intricate workings of set theory and
its role in the realm of computer science and programming.
A prime example of set theory in action is demonstrated by relational databases.
Within these databases, tables are utilized assets that consist of a unique array of rows.
These tables can be combined in numerous ways through set theory. For instance, if
you wish to select records that match on both tables, an inner join, also referred to as
an intersection, would be employed. Conversely, if you aim to obtain all matching
records from both tables, a full outer join, or a union in set theory, would be utilized.
When incorporating left and right joins, these concepts come together. By merging an
intersection with the difference between the two sets, a new set is formed that
encompasses solely the elements common to both sets.
It is crucial to understand that the above-mentioned structures are finite in terms of
element count, falling under the purview of discrete math. In contrast, continuous
math, such as geometry and calculus, deals with real numbers and approximations.
Set theory plays a vital role in the sphere of computer science and programming as it
enables the organization and manipulation of data. With an understanding of set
theory, one can execute various operations on sets of data, including sorting, filtering,
and aggregating.
6. Linear algebra
Linear algebra is a branch of mathematics that is necessary for understanding
computer graphics and deep neural networks. To comprehend it, you must first
understand three essential terms: scalar, vector, and matrix.
A scalar is a single numeric value. On the other hand, a vector is a list of numbers,
comparable to a one-dimensional array. Lastly, a matrix is like a grid or twodimensional array that contains rows and columns of numeric values.
What makes vectors and matrices fascinating is their ability to represent points,
directions, and transformations. Vectors can represent points and directions in a
three-dimensional space, while matrices can represent transformations that occur to
these vectors.
Consider this: when you move a player around in a video game, the lighting and
shadows, as well as the graphics, change magically. But it’s not magic; it’s linear algebra
that’s being distributed on your GPU.
Linear transformations can be a complex problem but let us examine a specific
scenario. Consider a 2D vector that instantly resides at 0.2X 3Y, denoting a solitary
point in an image. The application of scaling, translation, rotation, shear, and other
linear transformations is a possible avenue to explore. However, assume the intent is
to alter the point to be at 4X 6Y. In that case, a matrix representation of the scaling
factors is required, accompanied by the modification of the pinpoint to a column
vector. Through the use of matrix multiplication, the point can be shifted to its desired
location.
One area that utilizes this kind of operation extensively is cryptography, more
particularly RSA encryption. Moreover, deep neural networks are dependent on
vectors as a means of representing features. Through matrix multiplication, data can
be propagated between nodes of the network. Despite the apparent simplicity of the
math behind these transformations, the computing power required to handle the
massive amount of data is staggering. The study of math is integral to the
understanding of the magic that computers perform.
7. Combinatorics
Combinatorics is the art of counting and understanding patterns. It deals with the
combinations and permutations of sets and is used in a variety of applications, from
app algorithms to global databases. In this article, we will explore the concept of
combinatorics and its relevance in various fields.
Combinatorics is all about counting things, especially when combining sets into
combinations or permutations. For instance, if you have an app like Tinder, you might
need to count all the possible combinations of matches as part of a more complex
algorithm that figures out what to show the end user. Similarly, if you’re building a
globally distributed database, you need to figure out how many database partitions
you’ll need about the world. Ultimately, it’s all about understanding how patterns can
emerge.
The Fibonacci sequence is an excellent example of how combinatorics works. The
formula for the Fibonacci sequence can be seen in the image below. Writing a function
that generates it is a great exercise that allows you to grasp the concept of
combinatorics. Engineers at Google use the same kind of logic to render tiles on tools
like Google Maps. The Fibonacci sequence is also prevalent, and we can see it in the
spirals of a nautilus shell or the arrangement of sunflower seeds.
Combinatorics is closely related to graph theory. Graph theory is concerned with the
research of graphs, which are mathematical models applied to explain pairwise
relationships between things. Graph theory is widely used in computer science,
telecommunications, and social network analysis.
8. Statistic
AI is an area of study that heavily relies on the principles of statistics. This field deals
with developing algorithms and models that can learn from data and make predictions
or decisions based on that information. In essence, machine learning is a sophisticated
way of performing statistical analyses.
This piece aims to expound on the importance of statistics in the realm of artificial
intelligence and outline some fundamental statistical concepts that are crucial for
anyone seeking to pursue a career in AI. When you interact with an AI-powered
chatbot like ChatGPT, the system generates a response by computing the probability
that it fits the input prompt based on statistical models.
It is essential to have a clear grasp of the measures of central tendency, namely mean,
median, and mode, which are necessary statistical tools. Additionally, understanding
the concept of standard deviation is crucial as it helps in researching how close or far
apart values in a data set are from the average value.
Linear and logistic regression are two critical statistical methods that are commonly
applied in AI. In linear regression, the goal is to predict continuous values, such as the
loss or gain in monetary terms. The correlation between input and output variables is
linear, and the primary objective is to find the line that best fits the dataset.
On the other hand, logistic regression is valid in different problem scenarios such as
classification, for example, distinguishing between images of hot dogs and non-hot
dogs. The relationship between the input and output variables in this case is not linear
but rather a sigmoid function that predicts the probability of a Boolean variable being
true or false.
9. Complexity Theory
Complexity theory allows us to apprehend how much time and memory an algorithm
should ideally use. Big O notation is utilized in complexity theory to denote complexity.
Concerning time complexity, the input represents the number of steps needed to
execute a given task. For example, extracting a single element from an array would
give us 0 of 1, which is constant time, exceedingly swift, and straightforward.
Conversely, traversing an array would give us 0 of n, where n is the array’s length. If, in
each iteration of that loop, we traverse the same array once more, it would yield 0 of N
squared, which is significantly less efficient.
Sophisticated algorithms like binary search, halve the search space with each iteration,
resulting in logarithmic time complexity. Understanding the measurement of
complexity is fundamental for technical interviews and a necessary consideration in
general.
When attempting to comprehend complexity theory, it is imperative to understand
time complexity. The time complexity of an algorithm denotes the number of steps
required to complete a task. The greater the number of steps an algorithm takes, the
more time and memory it consumes. Thus, it is crucial to fathom how to measure
complexity in both time and memory.
Big O notation is frequently used to express temporal complexity. It is a mathematical
notation that defines the upper bound of a function’s growth rate. The symbol O(1)
denotes constant time complexity, which suggests that the method takes an identical
amount of time regardless of input size. In contrast, O(n) denotes linear time
complexity, implying that the algorithm’s runtime increases linearly with the input
size. Similarly, O(N²) represents quadratic time complexity, meaning that the
algorithm’s runtime advances exponentially with the input size.
For instance, if we need to traverse an array of size n, the time complexity would be
O(n). Nevertheless, if we cross the same display once more in each iteration of that
loop, the time complexity would be O(n²), which is considerably less efficient.
Binary search is a more sophisticated algorithm that halves the search area with each
iteration, resulting in logarithmic time complexity. It is vital to apprehend the tradeoffs between the time complexities of different algorithms to select the most efficient
one for a given task.
10. Graph Theory
When it comes to data structures, a graph is one of the most powerful and versatile
tools available. Graphs are used to represent complex relationships between objects or
concepts and can be found in everything from social networks to transportation
systems. In this article, we’ll take a closer look at what a graph is, and how it can be
used in programming and data analysis.
At its most basic level, a graph is composed of two types of elements: nodes (also
known as vertices) and edges. Nodes are the individual elements within the graph,
while edges are the connections between those elements. For example, in a social
network, a person might be a node, and the relationship between two people might be
an edge. If you love your mom, and your mom loves you back, this would be an
example of an undirected graph, because the relationship goes both ways. On the other
hand, if you love your OnlyFans girlfriend, but she doesn’t love you back, this would be
a directed graph because the relationship only goes one way.
Edges can also be weighted, meaning that one relationship is more important than
another. For example, in a transportation network, some roads might be faster or more
efficient than others, and these relationships could be represented by weighted edges.
Ascertaining whether a graph is cyclic or acyclic is a significant factor in graph theory.
If a node can traverse back to itself, then it’s considered cyclic, otherwise, it is referred
to as acyclic. This differentiation carries great significance since it impacts the
algorithms that can be employed to scrutinize or manipulate the graph.
As a software developer, constructing graphs from the ground up is a commonplace
activity. However, what’s even more critical is understanding how to traverse them.
Traversal denotes the procedure of exploring the graph to locate particular nodes or
relationships. Several algorithms can be utilized for graph traversal, but one of the
most prevalent approaches is Dijkstra’s algorithm. This method is operated to discover
the most efficient route through a graph.
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