Understanding Binary in Computing Systems

Prelims

When we talk about the computers we use every day—from smartphones and laptops to the servers running Nairobi’s stock exchanges—at their core, they all think in one language: binary. It’s a curious fact that despite all the complexity on our screens, everything boils down to just two symbols: 0 and 1. This fundamental simplicity behind binary implementation is what powers the digital revolution.

In this article, we'll look at how binary numbers are not just abstract math concepts but the nuts and bolts of storage, computation, and even programming logic. Knowing this inside out is especially useful for traders, investors, brokers, analysts, and entrepreneurs who rely increasingly on tech-driven tools. Understanding binary is no longer for just computer engineers; it gives you clearer insight into how data flows, gets processed, and influences the markets and business decisions.

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We'll cover key elements like:

• How binary numbers are represented and stored in modern computer systems

• The different data types and how they relate to binary

• Arithmetic operations conducted in binary and what that means for processing speed and accuracy

• Real-world examples of binary applications in Kenyan business and tech infrastructure

Grasping binary implementation offers not just technical knowledge, but a sharper lens through which to see the technology shaping Kenya's economic landscape today. Whether you’re tweaking a trading algorithm or evaluating hardware options, this understanding grounds you in the digital basics that matter.

Let’s get started by breaking down the building blocks of the binary system.

Basics of Binary Number Systems

Binary number systems form the backbone of how computers process and store information. At the core, understanding binary is essential for anyone dealing with technology—from traders using complex algorithms, to entrepreneurs setting up digital platforms. Unlike our usual decimal system, binary uses only two digits, 0 and 1, making it simpler but also uniquely suited for electronic circuits.

Knowing the basics of binary numbers can shed light on the efficiency behind data processing and storage. For example, when you check stock prices on a trading platform, the numbers displayed have been converted and handled internally in binary format. So, mastering this foundation isn’t just academic—it directly impacts how we design better software and hardware.

What Is a Binary Number?

A binary number is a sequence composed exclusively of 0s and 1s. Each digit in this sequence is called a bit. The position of each bit determines its value, much like how in the decimal system the position of each number represents powers of 10. In binary, those positions represent powers of 2.

Take the binary number 1011 as an example:

• The rightmost bit is 1, representing 2^0 (which equals 1)

• Next left is 1, representing 2^1 (which equals 2)

• Then 0 corresponds to 2^2 (which equals 0 in this case)

• The leftmost 1 corresponds to 2^3 (which equals 8)

So adding those up: 8 + 0 + 2 + 1 = 11 (decimal). This simple method allows computers to store any number efficiently by combining bits.

How Binary Differs from Decimal Systems

The decimal system uses ten digits (0 to 9), while binary sticks strictly to two. This difference might seem minor but is hugely impactful when it comes to machine operations.

Why two digits? Electronic components like transistors inside computers have only two states: on or off. Binary maps perfectly to this setup, with 1 meaning "on" and 0 meaning "off." Consider it like a light switch—either flipped up or down, no in-between states.

In contrast, decimal numbers are great for humans because they match with everyday counting and math but are complicated for circuits to interpret directly. For instance, representing the decimal number 9 requires multiple bits in binary (1001). So, while humans prefer base 10 for ease of use, computers prefer base 2 for reliability and simplicity.

Understanding this difference is key to appreciating how digital devices manage vast amounts of data swiftly and accurately.

Grasping the basics of binary numbers builds a strong platform for diving deeper into how computing hardware and software operate, which helps investors and entrepreneurs alike make informed decisions regarding tech-utilization and development.

Representing Binary Data in Computers

Computers don’t store information the way we do in notebooks or smartphones—they rely entirely on binary data, which boils down to just two states: 0 and 1. Understanding how binary data is represented inside a computer is key to grasping everything from how software runs on hardware to how data travels across networks. This section breaks down the nuts and bolts, showing how bits and bytes make up all the digital stuff we use daily.

Bits and Bytes Explained

A bit is the smallest unit of data, representing either a 0 or a 1. But dealing with single bits would be a nightmare—imagine trying to handle thousands of individual bits for everything. That’s why bits are grouped into bytes, with 1 byte equal to 8 bits. Think of a byte as a small bucket holding eight bits together to represent bigger chunks of information, like a letter, number, or symbol.

For instance, the ASCII code for the letter ‘A’ is 65 in decimal, which is 01000001 in binary—exactly one byte. By combining bytes, computers can represent complex data like images, videos, or even text in Swahili seamlessly.

Remember, everything you type, click, or view on a computer boils down to these tiny units of two possible values working together 24/7.

Storage of Binary Data in Memory

RAM and ROM Storage Methods

Memory in computers chiefly comes in two basic forms: RAM (Random Access Memory) and ROM (Read-Only Memory). RAM is the computer’s short-term memory—fast, flexible, and volatile, meaning it loses its data when power switches off. It's where active programs and data reside during operations. For traders running multiple charts or analytics tools, RAM ensures quick data access without lag.

On the flipside, ROM is non-volatile; it holds essential instructions like the boot-up sequence that turns your computer on. The binary data in ROM is fixed during manufacturing and helps your system wake up properly.

Both RAM and ROM rely on storing data as binary, often using transistor states represented by 0s and 1s. Modern RAM chips can store billions of bits quickly, while ROM chips ensure the system’s foundation is solid and secure.

Binary File Formats

When you save files—be it an investment spreadsheet or a market data chart—they get stored in various binary formats. Unlike plain text files, binary file formats pack data tightly to save space and speed up access.

Take Excel’s .xlsx files, for instance. They follow a binary structure that stores numbers, formatting, and formulas compactly. Images saved as .png or .jpg also use binary encoding, compressing the pixel data to reduce file size without dropping quality significantly.

Using binary formats ensures efficient storage and quick retrieval, which is a boon for entrepreneurs handling large datasets daily. It helps avoid bloated files and slow loading times, crucial when time literally means money on the market floor.

In the next sections, we'll explore how binary arithmetic and logic underlie processing and decision-making inside computers, powering everything from simple calculations to complex trading algorithms.

Binary Arithmetic and Logic Operations

Binary arithmetic and logic operations lie at the heart of how computers process information. Every calculation and decision a computer makes boils down to manipulating bits—those zeros and ones. For traders, investors, and entrepreneurs relying on data speed and accuracy, understanding these operations sheds light on how the tech underpinnings work behind the scenes, especially in financial software or trading algorithms.

At its core, binary arithmetic covers adding, subtracting, multiplying, and dividing numbers represented in base-2. Logic operations involve basic decisions computers make, such as comparing or combining bits using logic gates. These operations enable complex computations and control flow that power everything from real-time data analysis to secure transactions.

Basic Binary Arithmetic

Addition and Subtraction

Adding and subtracting in binary looks simpler but follows strict rules. Like decimal addition, if bits sum to more than 1, the carry goes to the next left bit. For example, adding 1 + 1 in binary results in 10 (which is 2 in decimal), carrying the 1. This carry-over concept is crucial in circuits like the Arithmetic Logic Unit (ALU) inside processors.

Subtraction often uses a method called two's complement, where the number to be subtracted is represented in a special binary form, letting the processor subtract by addition. This streamlines hardware design and speeds up calculations.

These arithmetic basics impact everything from how quickly stock prices update to how efficiently databases crunch numbers. As an example, in high-frequency trading, the ability to perform binary addition and subtraction rapidly can shave off milliseconds, a big deal in market movements.

Multiplication and Division

Binary multiplication resembles decimal multiplication but is more straightforward, using shifts and additions. Multiplying by 2 simply shifts bits one place to the left, a trick programmers tap into when optimizing code.

Division is a bit trickier and often handled by iterative algorithms inside CPUs. It involves repeated subtraction and shifting to produce quotient and remainder.

In daily terms, this means that financial modeling software can quickly compute complex formulas because these binary operations run at machine speed. Efficient multiplication and division underpin real-time risk assessment and portfolio simulations.

Logic Gates and Their Role in Binary Operations

AND, OR, NOT Gates

Logic gates are the basic building blocks allowing computers to make decisions based on binary inputs. An AND gate outputs 1 only if all inputs are 1; picture it as a strict traffic light letting cars pass only when all conditions are green.

An OR gate outputs 1 if any input is 1, acting like an easy-going gate letting through any car that arrives.

A NOT gate flips the input, turning 1 into 0 and vice versa, like a switch that reverses the signal.

These gates form the foundation of all digital circuits, including those managing transactions or encryption in fintech applications.

Combining Gates for Complex Functions

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By combining these simple gates, designers create more complex functions like multiplexers, adders, and flip-flops. For example, a half-adder circuit uses AND and XOR (exclusive OR) gates to add two bits and provide the sum and carry bits.

Such combinations handle decision-making and memory storage in processors. Without them, there wouldn't be any reliable computing devices.

In short, logic gates translate decision-making into a form machines understand – a cornerstone for any tech-driven business relying on computer-dependent systems.

Understanding how binary arithmetic and logic gates work can give professionals a clearer appreciation of why computers behave the way they do, and how software performance is ultimately tied to these fundamental operations.

Data Types Based on Binary Representation

Understanding data types as they relate to binary is key to grasping how computers process information. Data isn’t just a jumble of zeros and ones; it’s organized into types that tell the computer how to interpret those patterns. For traders, investors, and entrepreneurs relying on software, knowing these types sheds light on how data accuracy and performance come about beneath the surface.

In a binary system, different data types dictate everything—from simple numbers to complex financial models. This section breaks down two main categories: integers, which handle whole numbers, and floating-point numbers, which manage fractional values and very large or small quantities. Appreciating these distinctions helps in choosing the right data format when developing or analyzing financial software that requires precision and efficiency.

Integer Representation

Integers are the backbone of most computations and are stored as binary sequences. But not all integers are created equal: they can be signed or unsigned.

• Signed integers can represent both positive and negative whole numbers. They allocate one bit—usually the leftmost—to indicate the sign (0 for positive, 1 for negative). This is essential when your calculations involve losses or debts, like representing a drop in stock value.

• Unsigned integers only handle zero and positive numbers, thus maximizing the range of values it can represent for data that inherently can’t be negative, such as counts of transactions or inventory levels.

Practical Tip: If your software tracks cash flow that could dip below zero, signed integers are necessary to avoid errors. Conversely, unsigned integers are better when dealing with quantities that logically can’t be negative, making your programs leaner and faster.

Two's Complement

Two’s complement is the go-to method for storing signed integers in binary. It cleverly solves the problem of representing negative numbers in a way that allows simple arithmetic operations. Here's how it works:

• To find the two’s complement of a number, you invert the bits and add one.

• This allows the computer to perform subtraction using addition circuits, so hardware stays efficient.

For example, in an 8-bit system, the number -5 is stored as 11111011.

This method simplifies hardware design and speeds up calculations, making it fundamental in trading applications where quick numerical computations are routine. Two’s complement also helps avoid ambiguity with zero, ensuring there’s only one representation for it, unlike older methods.

Floating-Point Representation

When dealing with very large numbers, tiny fractions, or the in-between values that integers can’t cover, floating-point representation steps in. This is particularly relevant for financial analysis where precise interest rates, currency conversions, or risk measures are modeled with decimals.

IEEE Standard

Most computers use the IEEE 754 standard to represent floating-point numbers. This standard defines how numbers are stored in binary, splitting the number into three parts:

1. Sign bit: Indicates positive or negative.

2. Exponent: Represents the scale or magnitude.

3. Mantissa (or significand): Holds the actual digits.

This standardization means financial software can rely on predictable formats for calculations, no matter where it runs.

Here’s a brief example:

• The number 12.75 in IEEE 754 single precision (32-bit) would be split into those parts and then combined into the binary sequence stored in memory.

In practice, most trading platforms and financial libraries use IEEE 754 to ensure consistent calculations that underpin decision-making.

Precision and Range

The precision of a floating-point number refers to how many digits it can accurately represent, while the range indicates how big or small those numbers can get. However, floating-point can’t represent every decimal exactly, which sometimes leads to rounding errors.

For instance, a calculation for compound interest might lose or gain tiny amounts due to floating-point precision limitations. This subtlety is crucial when building models or software where accuracy affects cash flow or risk estimations.

• Single precision floats (32 bits) offer about 7 decimal digits of accuracy and can represent values roughly between 1.4×10⁻⁴⁵ and 3.4×10³⁸.

• Double precision (64 bits) offers about 15 decimal digits and a much wider range.

Choosing between these types depends on the application's need—where high accuracy or large ranges are necessary, double precision is better despite its bigger memory footprint.

In summary, understanding how integer and floating-point data types are represented in binary—and their benefits or pitfalls—offers essential insight for anyone involved in software that handles financial or numerical data. It ensures informed decisions on data structures can be made, leading to more reliable and efficient computing solutions.

Binary Implementation in Programming Languages

Binary implementation is the backbone of programming languages, whether high-level or low-level. It determines how computers process instructions, manipulate data, and communicate with hardware. For anyone involved in tech—in Kenya or elsewhere—getting a grip on this topic means better understanding how code runs under the hood, which ultimately leads to writing more efficient software and debugging faster.

How High-Level Languages Handle Binary

High-level languages like Python, Java, or C++ don’t directly expose binary operations in their everyday syntax. Yet, they provide tools to access and manipulate binary data, essential for tasks like encryption, compression, or device communication.

Bitwise Operators

Bitwise operators work at the binary level, dealing directly with the bits of a number. They're practical for performance-critical applications and systems programming. These operators include AND (&), OR (|), XOR (^), NOT (~), left shift (``), and right shift (>>).

For example, in financial trading algorithms that rely heavily on fast calculations or encoding flags in compact forms, bitwise operations help speed things up. The operator & can efficiently check if a particular flag bit is set, without the overhead of more complex code.

Memory Manipulation

Memory manipulation involves directly interacting with the way data is stored in memory, often through pointers or specific functions that let you read or write bytes or blocks of data. In high-level languages, this is usually abstracted away, but languages like C and C++ provide tools to handle pointers and manipulate memory explicitly.

In applications where control over hardware is needed—like IoT devices being developed locally or embedded systems—this ability is invaluable. It allows developers to optimize memory usage, reduce overhead, or write device drivers. For example, reading sensor data byte-by-byte might require precise memory manipulation to ensure the values are correctly interpreted.

Examples in Common Programming Languages

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These languages are most closely tied to hardware, offering a transparent window into binary implementation. Bitwise operators in C/C++ are straightforward and fast, crucial in systems programming, embedded applications, and performance-critical software.

C and C++ allow programmers to work with raw memory through pointers, which is essential for tasks like implementing custom data structures or interfacing with hardware. To illustrate, a trader developing real-time stock analysis software might use C++ to optimize data processing speed by manipulating memory directly.

c

include stdio.h>

int main() unsigned int flags = 0b1010; // Binary literal if(flags & 0b0010) printf("Flag set\n"); return 0;

This short code snippet mimics the previous check, showing how bitwise operations are just as accessible in Python.

Understanding binary implementation in popular programming languages empowers you to write faster, more efficient programs. Whether optimizing trading algorithms or working on hardware interfaces, mastering these tools gives you an edge.

In summary, binary operations and memory manipulation remain vital parts of programming, even in languages that largely abstract them away. The choice depends on your application’s speed and control requirements, but knowing these fundamentals always pays off.

Hardware and Binary Implementation

Hardware forms the backbone of any computing system, and its role in implementing binary logic cannot be overstated. At the core, hardware components handle the binary data—0s and 1s—that software relies on. When you’re examining anything from a basic calculator to complex servers running Nairobi's stock market data feeds, understanding how hardware really processes binary is key to grasping overall system performance.

This section sheds light on how binary operations happen behind the scenes in physical components. Knowing this helps traders and tech entrepreneurs appreciate how computing speeds and efficiency might impact the apps or systems they depend on. For example, a lag in processing or data corruption could mean delayed trading signals or lost transactions, so robust binary-handling hardware matters.

Binary in Microprocessors

Instruction sets are basically the commands that a microprocessor understands and executes. Each set is a predefined collection of binary instructions telling the processor what to do next—like adding two numbers, shifting data, or storing a value in memory. Think of this as a language the chip speaks, made up entirely of binary commands.

Practical relevance? If you have ever heard of the x86 or ARM architectures in CPUs—which power most laptops and smartphones respectively—you’re already familiar with instruction sets. They define how breakthrough models like Intel’s Core or Qualcomm Snapdragon chips operate. Knowing this helps investors understand hardware limitations and potential upgrades that might affect your tech portfolio.

Register usage refers to how the microprocessor temporarily stores data during operations. Registers hold bits or bytes for immediate access, kind of like a workspace on a trader’s desk—quick, limited, but essential for multitasking. Efficient register management lets chips perform many calculations without resorting to slower memory.

For example, a 64-bit processor has registers that can process 64 bits of data at once, boosting computing power drastically compared to a 32-bit one. Getting familiar with register sizes and their usage explains why some devices perform faster or handle complex software better.

Digital Circuits Using Binary

Flip-flops and counters are fundamental circuit components that create memory and timing in digital systems. Flip-flops are bistable devices, meaning they hold a single bit of data (either 0 or 1) until switched. Counters use flip-flops to keep track of sequences, such as counting clock pulses or events.

In practical terms, these circuits are behind timers in embedded systems, frequency counters in radios, or even in automated Nairobi traffic lights control. Their reliability is critical for processes where exact timing or sequence matters.

Multiplexers and decoders serve as selectors and translators in digital logic. A multiplexer chooses one binary input out of many based on a control signal, streamlining data flow by reducing wiring complexity. Conversely, a decoder takes binary input and activates a specific output line, essential in displaying data on screens or directing signals.

For instance, in financial kiosks across Kenyan markets, multiplexers might manage multiple customer inputs, while decoders ensure the right information appears on screens. Understanding these components uncovers how simple binary decisions escalate into complex system behaviors.

Hardware and binary logic form the foundation for all computing tasks — from quick number crunching essential for trading platforms to controlling automated billing systems across Kenya. Grasping these basics ensures better insight into the tech powering your business.

This hardware insight bridges the gap between abstract binary data and practical technology deployed daily, making it crucial knowledge for anyone involved in tech-centric industries or investments.

Challenges in Binary Implementation

Binary systems lie at the core of all computing, but they come with their own set of hurdles that every developer, hardware engineer, and systems analyst must grapple with. Understanding these challenges isn't just academic—it directly affects system reliability, performance, and accuracy. The two main issues often encountered are overflow and underflow in computations, alongside precision difficulties particularly with floating-point numbers. Predicting and mitigating these risks helps traders, entrepreneurs, and brokers manage data safely and avoid critical errors in financial software or trading platforms.

Handling Overflow and Underflow

Overflow happens when a calculation produces a number too large for the system to represent within its allotted number of bits. For instance, consider an 8-bit system where the maximum unsigned value it can store is 255 (11111111 in binary). Adding any amount above this limit without proper handling causes the number to "wrap around" to zero or a smaller value, leading to serious computational errors.

Underflow, on the other hand, occurs when numbers near zero become too small to be represented, effectively getting treated as zero. This can cripple calculations in financial modeling where tiny fractions matter.

Practical example: In trading algorithms that rely on precise price increments, overflow can cause a sudden drop in reported prices, or underflow can erase tiny gains, throwing off profit calculations. Some programming languages tackle this with built-in exceptions or special checks. For example, C++ offers safeguards like overflow-checked arithmetic functions. Still, engineers must explicitly code for such checks, especially during high-frequency trading where errors can't be tolerated.

Precision Issues in Floating-Point

Floating-point arithmetic, standardized under IEEE 754, tries to balance range and precision. But it’s not perfect. Due to limited bits to store the mantissa and exponent, certain decimals can’t be stored exactly. This introduces tiny rounding errors that pile up in calculations.

Imagine a brokerage software that calculates compound interest daily for thousands of client accounts. Over time, these small rounding errors can add up and cause discrepancies amounting to significant financial differences. Another common example is currency conversion calculations where fractions of cents matter.

In practice, many financial software solutions use fixed-point arithmetic or arbitrary-precision libraries like GMP (GNU Multiple Precision Arithmetic Library) instead of floating-point. This avoids rounding errors but can slow down computations, a trade-off developers must weigh carefully.

Ignoring overflow, underflow, or floating-point precision can lead to flawed analysis and financial loss. The key lies in choosing the right numeric representation for your application's demands.

Understanding and addressing these challenges keeps software trustworthy, especially in sectors like finance and trading where binary calculations underpin real-world decisions.

Practical Applications of Binary Implementation

Understanding how binary is implemented beyond theory is key to seeing its real-world value, especially for those involved in trading, investing, and technology sectors in Kenya. Binary systems form the backbone of how computers communicate, store data, and perform computations. This section focuses on practical uses where binary implementation directly impacts efficiency, accuracy, and security — elements crucial to businesses and tech entrepreneurs alike.

Networking and Data Transmission

Binary Protocols

Networking depends heavily on binary protocols, which are sets of rules for communication between devices using binary data. These protocols ensure that information traveling through the internet or local networks is packaged, transmitted, and understood consistently. For example, the Transmission Control Protocol (TCP) uses binary headers to track packets and maintain communication stability. This is crucial in financial markets where even a tiny delay or data loss can cause major issues.

Binary protocols achieve efficiency by compressing instructions into compact binary formats, reducing size and speed delays. For the Kenyan business ecosystem, this means faster transaction speeds and smoother data flow, a critical need for high-frequency trading platforms or real-time analytics services.

Error Detection and Correction

Binary communication isn’t foolproof; noise or interference can flip bits and corrupt data. Error detection and correction methods are necessary safeguards. Techniques like parity bits, cyclic redundancy check (CRC), and Hamming codes help identify and fix errors without needing to resend data.

For example, when a payment goes through M-Pesa or another digital wallet, error-correcting protocols make sure the binary transaction data arrives intact. This avoids costly mistakes like wrong transfer amounts or failed payments, providing confidence for users and service providers alike.

Effective error handling in binary transmission not only protects data integrity but also builds trust in digital services.

Storage Devices

Hard Drives and SSDs

At their core, storage devices like hard drives (HDD) and solid-state drives (SSD) rely on binary to store and retrieve data. HDDs use magnetic polarity to represent bits — north or south poles correspond to 0s and 1s. SSDs, meanwhile, store charge in transistors to achieve the same binary representation but without moving parts, resulting in faster access and greater durability.

In Kenyan markets and offices, choosing between HDDs and SSDs can affect data handling speeds and system reliability, especially for startups relying on quick data access or backup. For instance, SSDs can reduce the time you spend waiting for reports or video processing, improving overall productivity.

Binary Encoding Methods

Binary encoding refers to how data is represented in these storage devices. Common methods include Non-Return-to-Zero (NRZ), Manchester encoding, and more sophisticated techniques like 8b/10b encoding used in SATA drives.

These encoding schemes ensure that binary sequences remain synchronized during read/write processes and help detect errors. By optimizing encoding, storage devices can minimize power consumption and maximize throughput, which is critical for businesses managing large databases or running cloud services.

In practice, understanding these encoding methods helps Kenyan tech leaders make better decisions about infrastructure — choosing hardware that balances cost, speed, and data integrity according to their specific needs.

To sum it all up, binary implementation in networking and storage is far from abstract. It underpins the reliability and speed of modern digital systems, which businesses in Kenya rely on daily. Whether it’s ensuring a smooth online banking transaction or selecting the right storage technology for a fintech startup, knowing the nitty-gritty of binary applications can provide a solid edge.

Converting Between Binary and Other Number Systems

Understanding how to convert binary numbers to other number systems and back again is a key skill in computing. It’s not just an academic exercise—this forms the backbone of how computers interpret and present data we can understand, like prices, stock values, or transaction IDs. Without these conversions, communicating data between software, hardware, and humans would be a tough nut to crack.

At its core, this conversion lets us translate binary, which uses just 0s and 1s, into other numeric formats humans use daily, particularly decimal and hexadecimal. This ability is especially important in environments like Kenya where entrepreneurs and traders rely on computers to manage business data efficiently and accurately.

Binary to Decimal and Vice Versa

Conversion techniques

Switching between binary and decimal involves breaking down numbers into powers of two or ten. For example, to convert the binary number 1011 to decimal, you multiply each bit by 2 raised to the power of its position (starting from zero on the right). So, that’s (1×2³) + (0×2²) + (1×2¹) + (1×2⁰), which equals 8 + 0 + 2 + 1 = 11 in decimal.

Conversely, to turn a decimal number like 19 into binary, you repeatedly divide by 2, recording the remainder until you reach zero. The remainders, read backward, give you the binary equivalent 10011.

plaintext Decimal to Binary Example: 19 ÷ 2 = 9 remainder 1 9 ÷ 2 = 4 remainder 1 4 ÷ 2 = 2 remainder 0 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1

Reading remainders bottom-up: 10011