If I ask you to divide negative 5 by 2, what
is the remainder? Just think about that for a
second. Negative five divided by two. You'd probably
think there's, you know, one universally correct,
perfectly logical math answer to that question.
Right. Exactly. But if you take that exact same
math problem and hand it to a system built by
Apple and then hand it to a server running Microsoft
architecture and then, I don't know, ask a pure
mathematician, you are going to get completely
different answers. It really is wild. It completely
shatters this illusion we all have that math
inside a computer is this pristine absolute truth.
It totally does. And that's exactly what we're
getting into today. Welcome to the deep dive.
We are looking at an operation that you probably
learned in the fourth grade, assumed you fully
understood, and just haven't thought about since.
No, we're talking about the modulo operation.
Oh. Or often just called mod. Right, mod. And
it is secretly running the digital world you
interact with every single day. Our mission today
is to explore how a concept as basic as finding
the remainder of a division problem actually
breaks down in spectacular ways when it meets
negative numbers. And I mean, it isn't just an
academic work either. This breakdown has created
massive philosophical debates, split programming
languages into warring factions and caused incredibly
subtle system crashing bugs in the software we
all rely on. Yeah, the bugs are the crazy part.
Yeah. So whether you are a seasoned programmer
who deals with this daily or just someone listening
on a smartphone right now who wants to know how
the invisible digital gears actually turn, this
is going to completely change how you view basic.
division. Absolutely. So let's start at the absolute
baseline, because we really have to build the
foundation before we can break it. What exactly
is modulo doing when we aren't dealing with negative
numbers? Okay, so at its core, modulo is just
an operation that asks for the remainder of a
division problem. You have a dividend, that's
the number you're dividing, and a divisor, which
we call the modulus. So if you run the expression
5 modulo 2, it evaluates to 1. Right, because
two goes into five twice, which gets you to four,
and then you have one left over. Exactly. Or,
you know, nine modulo three, that evaluates to
zero. Three goes into nine exactly three times.
There's nothing left over. Makes sense. So mathematically,
if you're doing a modulo operation with a positive
divisor, let's just call it n, the result is
always going to be locked in a specific range.
It'll always be zero up to n minus one. Okay,
so if I'm doing modulo 5, the only possible answers
in the universe are 0, 1, 2, 3, or 4. Right.
I mean, I can never have a remainder of 5, because
if I had 5 left over, my divisor of 5 would just
go into it one more time. Precisely. And in computing,
this entire operation is held together by one
strict, non -negotiable math equation. The dividend
equals the divisor times the quotient, plus the
remainder. Okay, let's make that concrete. The
pizza analogy always works best for me. Oh yeah,
go for it. If I have five leftover slices of
pizza, that's my dividend. I have two friends,
that's my divisor. How many slices do they each
get? That's the question. They each get two.
Right. So two times two is four. Exactly. Plus
the remainder. I have one slice left in the box.
So five equals two times two plus one. That is
the perfect clean elementary school version.
Yeah. But here is where the math hits a brick
wall. What happens when exactly one of those
numbers is negative? This is where I started
losing my mind reading the source material. How
do you calculate a remainder when someone owes
you negative pizza slices? I mean, what does
that even mean? Well, let's do the math out loud
using that exact formula. The formula is, again,
dividend equals divisor times quotient plus remainder.
OK, I'm with you. Let's make the dividend negative
5 and the divisor positive 2. Negative 5 divided
by 2. We need to find the quotient and the remainder.
Okay, so negative 5 equals 2 times something
plus a remainder. Right. And here is the trap.
Because we are dealing with integers, you know,
whole numbers, there are actually two completely
valid ways to solve this equation. Wait, two
valid ways? Yeah. Path A... You could decide
the quotient is negative 2. 2 times negative
2 is negative 4. Okay. And to get from negative
4 down to negative 5, your remainder has to be
negative 1. Let me track that. So negative 5
equals 2 times negative 2, which is negative
4 minus 1. Yeah, the math works perfectly. The
remainder is negative 1. But look at path B.
What if you decide the quotient is negative 3?
Oh, interesting. Right, because 2 times negative
3 is negative 6. And to get from negative 6 up
to negative 5, your remainder has to be positive
1. Wait, let me think. Negative 5 equals 2 times
negative 3, which is negative 6 plus 1. Wow,
the math works perfectly there too. So the remainder
is positive 1. You literally have two completely
correct mathematical answers for a basic division
problem. Yes. The choice of remainder entirely
depends on which of two consecutive integers
you pick for the quotient. Do you pick negative
2 or negative 3? That is wild! And in pure number
theory, mathematicians saw this ambiguity and
just made a rule. They agreed to always choose
the positive remainder. They wanted remainders
to behave like distances, always zero or greater.
But I'm guessing computer scientists didn't play
along with the mathematicians. Not at all. Because
hardware engineers and software developers have
to actually build systems that store and process
these numbers efficiently. They didn't just disagree
with the mathematicians, they disagreed with
each other. Oh, of course they did. Right. This
ambiguity forced the invention of different computational
rules, which basically split the programming
world into distinct tribes. The tribal war is
of modulo. I love this part. So break down these
factions for me. If I'm writing code, whose rules
am I actually following? Well, the first major
tribe uses what is called truncated division.
This was heavily favored by early hardware architects.
In truncated division, when you calculate the
quotient, you always round towards zero. Round
towards zero. OK. Think of a number line. If
the exact decimal answer is negative 2 .5, truncated
division just chops off the decimal and rounds
towards zero, giving you a quotient of negative
2. Right, and as we just proved with Path A earlier,
if your quotient is negative 2, your remainder
has to be negative 1. Exactly. Because of how
rounding towards zero works out mathematically,
the remainder in truncated division will always
take the exact same sign as the dividend. OK,
so if the number you are dividing is negative,
your remainder will be negative. You got it.
And this tribe includes massive foundational
languages like C, C++An, and Java. OK, so that's
tribe one, the hardware purists, I guess. What's
the second tribe? The second major tribe uses
floor division. This is championed by Donald
Knuth, who is one of the legendary figures in
computer science. Yeah, definitely. Floor division
doesn't round towards zero. It always rounds
down toward negative infinity. So on a number
line, negative 2 .5 rounds down to negative 3.
Which takes us to path B. If the quotient is
negative 3, the remainder has to be positive
1. Right. In floor division, the remainder always
takes the same sign as the divisor. If you are
dividing by a positive two, your remainder will
always be positive, no matter what number you
started with. That makes a lot of sense. Yeah.
And this approach was adopted by languages that
were designed to be more mathematically intuitive,
like Python and Ruby. OK. And just to be thorough,
is there a third tribe? There is. Euclidean division,
promoted by computer scientist Raymond T. Boot.
He looked at both truncated and floored division
and argued they were both flawed. Why? Because
they allowed rematers to flip signs wildly, depending
on the inputs. He argued that systems should
force the remater to always be non -negative,
zero or greater, period. just like the pure mathematicians
wanted. Okay, so stepping back, C++ is doing
truncated division. Python is doing floor division.
This means if I have a massive tech stack, say
I've got a backend server written in C++ A, and
it's passing data to a machine learning script
written in Python, they could process the exact
same negative number with the exact same modulo
operation and spit out two totally different
results. Yes, and they will do it silently. The
computer won't throw an error. It just assumes
you, the programmer, know which tribe you are
operating in. That is terrifying. How does the
digital world even function if the foundational
languages can't agree on what a remainder is?
It feels like we build modern civilization on
a fault line. It really does. And it leads to
catastrophic, yet incredibly subtle, bugs. Let's
look at the most famous pitfall from the source.
It's something pretty much every programmer encounters,
usually the hard way. I want to hear this, because
up until now, this has felt a bit like an esoteric
math debate. How does this actually break real
-world code? Okay, imagine you were writing code
for a bank or, I don't know, a video game, and
you need to test if a number is odd or even.
Okay, that is literally computer science 101.
It's the first logic test anyone writes. You
take your variable, let's call it n, and you
do n modulo 2. If the answer is 1, the number
is odd, because any odd number divided by 2 leaves
a remainder of 1. It seems bulletproof, right?
2 is 1. 7 mod 2 is 1. So you write the code,
if n mod 2 equals 1, then trigger the logic for
an odd number. But wait, we just established
that if n is negative 5, and I'm writing this
in C++ or Java, those languages belong to the
truncated division tribe. The remainder takes
the sign of the dividend. Exactly. Your dividend
is negative 5. So negative 5 modulo 2 doesn't
return 1. It returns negative 1. And what does
your code say? Your code specifically checks
if the result is exactly equal to positive 1.
Oh, wow. So nmod 2 equals 1 evaluates to false.
Yes. The system looks at negative 5, sees the
remainder is negative 1, and incorrectly decides
that negative 5 is not an odd number. That is
a massive logic failure. Think about a financial
algorithm flagging every odd -numbered transaction
for audit, but it silently skips all the refunds
or debts because they are negative numbers. Exactly.
The logic just silently bypasses the odd numbers.
To fix this, programmers have to train themselves
out of their intuitive math habits. The mathematically
safe way to check if a number is odd across any
language is to test if the remainder does not
equal zero. Because whether the system spits
out a positive one or an negative one, neither
of them is zero. Right. And mod two does not
equal zero. Zero doesn't have a sign. Remainger
zero means the number is perfectly divisible
by two. So it completely bypasses the tribal
wars of negative remaingers. That is brilliant,
but also incredibly frustrating that we have
to work around it. Okay, getting the math right
is obviously a minefield. But in computing, getting
it right isn't enough. It also has to be blisteringly
fast. And division, which Modulo relies on, is
famously slow, right? Oh, very slow. For a computer's
central processing unit, operations like addition,
subtraction, or shifting bits around take barely
any effort, maybe one clock cycle. But division
is computationally expensive. It requires the
processor to grind through multiple steps, taking
20, 30, or even 40 clock cycles to calculate
a quotient and a remainder. Wow. So if I'm playing
a video game rendering at 60 frames per second,
and the physics engine is running millions of
modulo operations to track bouncing particles
or screen wraparounds, that division becomes
a massive bottleneck. The game would stutter.
How do systems speed this up? Hardware engineers
and compiler writers devised a brilliant shortcut,
but it only works for a special case when your
divisor is a power of two. Things like 2, 4,
8, 16, 32. Okay, powers of two. When you were
doing modulo with a power of 2, you can completely
skip the division and replace it with a lightning
-fast operation called a bitwise AND. Okay, I
have an analogy for this, but I need to make
sure I'm mapping it right. If I asked you to
divide a massive number, let's say 8 ,432 ,567
by 10, and tell me the remainder, you wouldn't
pull out a pencil and do long division. No, I
would just look the last digit, the 7. Right.
Because our human number system is base 10, so
dividing by 10 or 100 or 1000 and finding the
remainder just means looking at the end of the
number. The digits themselves already give you
the answer. Does the computer do the same thing?
That is exactly what the computer is doing. But
in base 2, computers think in binary 1s and 0s.
Every position in a binary number represents
a power of 2. Oh, I see where this is going.
So if you ask a computer to calculate modulo
8, it doesn't need to do long division? 8 is
a power of 2, the computer just needs to look
at the last three digits of the binary numbers.
Because anything beyond the last three digits
is a multiple of 8. So how does the bitwise NND
operation actually do that? A bitwise NND literally
acts like a physical mask. You take your massive
tinary number and you apply a mask that says
force all the higher bits to zero and only keep
the bits at the very end. That's so cool. It
happens instantaneously at the hardware level.
No math required, just a clean chop. The formula
is x modulo 2 to the power of n equals xand2
to the power of n minus 1. Okay, so if I want
to do modulo 4, the compiler secretly rerates
my code to say xand3. It just masks out everything
except the last two bits. That is so elegant.
It is a massive performance hack. Modern compilers,
the software that translates your human readable
code into raw machine instructions, will actively
hunt through your code for this. Really? If they
spot you doing modulo with the power of two,
they will automatically swap out your slow division
for this fast bitwise A &D operation. You get
the speed without having to write weird binary
logic yourself. But wait. I'm sensing a trap.
We just spent 10 minutes talking about the chaos
of negative numbers. How does this bitwise hack
handle negative numbers? You've hit the exact
issue. The compiler optimization fails completely
if you are using signed integers in a language
that uses truncated division, like C or C++AT.
Why? What breaks? Think about what a bitwise
AND operation actually does. It's looking at
raw bits. ones and zeros. It doesn't understand
the concept of a negative value. Oh, right. To
a bitwise operation, the sign of a number is
just another bit, usually the first bit in the
sequence. When you apply that mask to chop off
the higher bits, you are also stripping away
the sign bit. Oh. Which means a bitwise A &D
will always, 100 % of the time, spit out a positive
number. Exactly. But remember the rule of the
truncated tribe. If the dividend is negative,
the remainder must be negative. So if the compiler
tried to be helpful and secretly swapped my slow
modulo operation for a fast bitwise AND, a negative
dividend would suddenly start returning a positive
remainder. It would literally change the mathematical
output of my program just to save a few CPU cycles.
Precisely. It would corrupt the data. So to prevent
this, compilers for languages like C are forced
to abandon the simple, elegant bitwise AND if
there's any chance the variable could be negative.
That is so frustrating. Or they have to inject
a much more comp - sequence of bitwise ORs and
NOTs just to manually preserve that negative
sign which eats into the performance gains. It
is wild how an arbitrary decision made by hardware
architects decades ago to round towards zero
is actively preventing modern compilers from
optimizing code today. Okay, we have zoomed way,
way in on the microscopic level of binary bits.
Let's zoom out. Because understanding modulo
isn't just about avoiding coding bugs, it is
actually the foundational math that secures the
modern internet. Yes. When you elevate modulo
beyond basic arithmetic, you enter the realm
of cryptography. The idea here is that modulo
creates a wraparound effect. Modular arithmetic
is essentially clock math. Right. That is the
perfect way to visualize it. Think of a standard
12 -hour clock. If it is 10 .000 a .m. and I
add 5 hours, the time doesn't become 15 .00 o
'clock. Right. It hits 12 and wraps back around
to 3 .0 p .m. You just did modulo 12 in your
head. 10 plus 5 is 15. 15 modulo 12 leaves a
remainder of 3. Modular arithmetic forces numbers
to wrap around a fixed boundary over and over
again. So it just keeps spinning? Exactly. No
matter how infinitely large your starting number
is, the modular operation instantly snaps it
back onto the clock face. So how does snapping
a number onto a clock face secure my credit card
transactions? It creates a mathematical trapdoor,
a one -way function. In cryptography, protocols
like the Diffie -Hellman key exchange computers
take massive prime numbers and raise them to
massive powers. We're talking numbers with hundreds
of digits. Unimaginably huge. Right. Then they
apply a modulo operation using another massive
prime number as the modulus. So they are taking
a galaxy -sized number and wrapping it around
a galaxy -sized clock face. Yes, and here is
the trapdoor. If I tell you the final answer,
let's say I tell you the clock hand is currently
pointing at three, you have absolutely no way
of knowing how many times that hand spun around
the clock to get there. Did it spin around once?
A billion times. A trillion times. Oh wow. Because
the quotient is completely discarded? Modulo
only gives you the remainder? The history of
how you got there is erased? Exactly. It is mathematically
very easy for a computer to spin the clock forward
and find the remainder. But it is practically
impossible for a hacker to look at the remainder
and reverse engineer the original mass of numbers.
The information has been locked behind the modulo
operation. So the exact same mathematical quirk
that forces a college freshman to tear their
hair out over a negative one in a basic parity
test is the same foundational property that allows
humanity to securely transmit state secrets across
a public Wi -Fi network. It is the exact same
underlying mechanism, the destruction of the
quotient. That is incredible. Before we wrap,
there is one last real -world application I want
to touch on, the concept of modulo with offset.
because a clock starts at 12 or zero, but not
everything in life starts at zero. Right. Often
we need a cyclic pattern, but we need the boundary
shifted. We need the remainder to lie between
a specific offset, let's call it D, and D plus
N minus one. The classic example of this is a
calendar. Think about the days of the week. We
operate on a seven -day cycle, module of seven,
but we don't call Monday day zero and Sunday
day six. We number them one through seven. Exactly.
If you are a programmer, writing a date calculation
script, say, figuring out what day of the week
a bill is due, 45 days from now, a standard Modulo
7 operation will occasionally spit out a zero.
But there is no day zero on your calendar. Your
program will crash. Exactly. So you apply an
offset of 1. You adjust the formula, usually
incorporating the floor function, to shift the
mathematical boundaries so the output cleanly
wraps around from 1 to 7 instead of 0 to 6. That
makes total sense. It is a perfect example of
how pure math has to be wrestled and nudged to
actually mirror human reality. Which brings us
to the end of our deep dive. And honestly, my
brain is slightly spinning. We started with the
simplest concept imaginable, dividing leftover
pizza slices to find a remainder. But the exact
second we introduced the reality of negative
numbers, we uncovered a full -blown mathematical
crisis. Yeah, we navigated the tribal wars of
computing. The hardware purists using truncated
division, the mathematical idealists using floor
division, and the chaos of them disagreeing silently.
We saw how failing to understand those tribes
leads to the infamous negative one logic trap
that breaks even the simplest code. We dug into
the binary hardware to see how bitwise and Dmasks
can hyper optimize performance unless the ambiguity
of negative signs forces the compiler to abandon
the shortcut entirely. And finally, we saw how
the wraparound information destroying nature
of modulo forms the one way trap doors that make
modern cryptography possible. It really highlights
that in computer science, nothing is quite as
clean as it appears on a chalkboard. So I want
to leave you with the final thought to mull over.
We open by talking about the expectation of absolute
precision. We assume that when we type an equation
into a calculator, the answer we get is a universal
truth. But today we saw that something as fundamental
as dividing by 2 and finding a remainder doesn't
actually have a single objective answer. Right.
It has tribes. It has hardware constraints. It
has compromises. So the next time you look at
a seemingly basic rule or a standard interface
in the technology you use every single day, ask
yourself what edge case did the creators have
to argue over to make this work? If basic arithmetic
itself has to compromise to fit inside a computer
processor, what other absolute truths in our
digital world are really just tribal agreements
in disguise? It forces you to question the foundation
of every system you use. It really does. Thanks
for joining us on this deep dive. See you next
time.