Octal to Decimal

Introduction

The octal numeral system, characterized by its base-8 representation, often evokes curiosity among students, hobbyists, and professionals engaged in fields such as computer science, electronics, mathematics, and beyond. While less ubiquitous in everyday life than the decimal system, octal still possesses a significant historical and practical relevance, particularly in computing contexts. Essentially, converting Octal to Decimal entails translating numbers expressed in base-8 into their base-10 counterparts. This form of conversion is central in many engineering and technological environments, where specialized representations are more advantageous than conventional decimal formats.

Over the decades, octal has played a pivotal role in representing data in computers. Tracing its origins back to early computing architectures, engineers and programmers employed octal to compress binary notation into fewer digits, ensuring better readability and ease of use. Even though hexadecimal has largely overshadowed octal in modern programming paradigms, octal remains a significant method for denoting file permissions in Unix-based systems—an extremely common need for systems administrators. In these scenarios, knowledge of converting Octal to Decimal is sometimes essential to ensure accuracy in day-to-day tasks. Beyond systems administration, octal usage plows its way into embedded systems, low-level microcontroller programming, and various resource-limited computing platforms where data representation efficiency matters.

This article dives deep into octal’s legacy, practical examples of how to shift it into decimal form, and the scenarios where such conversions are used on a routine basis. You will come away from this lengthy exploration with a thorough understanding of why and how to convert octal numbers to decimal, different methods to accomplish the conversion, and the role of these bases in broader computing contexts.

Understanding Octal and Decimal Systems

Before we plunge into the intricacies of the Octal to Decimal conversion process, it’s important to have a foundational grasp of both the octal and decimal numeral systems themselves. Essentially, any numeral system can be defined by the number of unique digits it utilizes as the base for representing numbers.

1. Decimal System (Base-10)
Arguably the most recognizable number system worldwide is the decimal system. We use it day in, day out, so extensively that it can be easy to forget it is merely one type of representation: base-10. It uses the digits 0 through 9. In decimal, every digit’s place value represents a power of 10, determined by the position of the digit from right to left. It’s a system deeply integrated into global commerce, measurements, and standard arithmetic operations.

2. Octal System (Base-8)
Meanwhile, the octal system employs eight unique digits: 0 through 7. Each digit occupies a place value determined by powers of 8. For instance, consider the three-digit octal number 745 (octal). This number represents:

• 7 × 8² (because the 7 is in the hundreds place)
• 4 × 8¹ (the 4 is in the eights place)
• 5 × 8⁰ (the 5 is in the ones place)

Adding those products together yields its equivalent decimal value. We’ll delve deeply into specific step-by-step examples shortly, but the core mechanism revolves around the same principle: summing each digit multiplied by the base raised to that digit’s position (starting from 0 on the far right). Once you internalize this simple formula, you can reapply it to convert Octal to Decimal for any octal string, regardless of length.

Historical Significance of the Octal System in Computing

The significance of the octal system historically emanates from its strong ties to early computer architectures. In the mid to late 20th century, many computing systems used word sizes or instruction sets that aligned more neatly with base-8 than base-16. Consequently, representing memory addresses, machine instructions, or raw data in octal was common practice.

1. Early Computers and Octal
Some early machines had word lengths of 12, 24, or 36 bits—divisible by 3, which is a factor that fits perfectly with base-8 (each octal digit can directly map to 3 bits). Hexadecimal (base-16) maps neatly to groupings of 4 bits, so while modern 8-, 16-, 32-, or 64-bit architectures have shifted the momentum towards hexadecimal, octal overcame a significant barrier in older architectures by supplying a direct correlation to the underlying binary data. In such scenarios, converting data from Octal to Decimal provided a quick check on memory addresses or data segments.

2. Assembly Language and Octal
Assembly languages, especially for older or specialized hardware, often had developers referencing memory locations in octal. This was simply more intuitive when dealing with bit groupings of three. Whenever decimal values needed to be cross-referenced for arithmetic, developers were forced to convert from octal into decimal. Those adept at quick mental math could move between Octal to Decimal mentally, performing multiplication and addition calculations in their head. Although that skill might seem a bit esoteric now, it was crucial in certain computing circles.

While modern computing rarely pushes users to rely exclusively on octal, that doesn’t mean the practice is defunct. Instead, it has become a niche technique for certain tasks, especially involving Unix file permissions (rwx, chmod) in system administration, microcontrollers, or academic explorations into different numeral systems.

Why Convert Octal to Decimal?

It might be worth exploring why, in an age dominated by higher-level programming languages and advanced integrated development environments, one might still need to do an Octal to Decimal conversion.

1. File Permissions in Unix, as previously mentioned, still revolve around octal notation. A typical Unix (or Unix-like system) user might use commands like chmod 755 filename to set user and group permissions. Under the hood, that 755 is an octal representation telling the file system how to regulate read, write, and execute permissions. Interpreting which decimal number that octal corresponds to may sometimes be required.

2. Low-Level Computing in embedded systems or academic exercises still demands that kind of knowledge. Suppose you are debugging or reading a memory dump that has an octal representation. You’d likely be required to interpret or convert that to decimal to quickly ascertain the meaning behind it.

3. Standardized Reporting of certain measurements may rely on decimal. If your data source is spitting out octal data, but your final system or user expects it in decimal form, you’ll need a reliable method of conversion.

4. Educational Value stands on its own, as understanding and performing conversions across numeral systems fosters stronger mathematics and computer science fundamentals. A thorough mastery of the mechanics behind Octal to Decimal conversions is part of a well-rounded skill set, especially for students venturing into degrees in engineering, mathematics, informatics, or data science.

Basic Mathematics of Octal to Decimal Conversion

At the core of this process, the method to convert Octal to Decimal can be boiled down to a single formula or approach. Each octal digit is multiplied by 8 raised to a power corresponding to the digit’s position from the right, starting at 0. Summing all these products yields the decimal result. Let’s break this down with a simple example:

1. Start with an octal number, say 234 (octal).
2. Identify digit positions from right to left. The digits are:
   • 4 (rightmost, position 0)
   • 3 (middle, position 1)
   • 2 (leftmost, position 2)
3. Compute each digit multiplied by 8 raised to the digit’s position:
   • 2 × 8² = 2 × 64 = 128
   • 3 × 8¹ = 3 × 8 = 24
   • 4 × 8⁰ = 4 × 1 = 4
4. Sum the values: 128 + 24 + 4 = 156
5. So, 234 (octal) = 156 (decimal).

Whether the number has 2 digits or 20, the process is effectively the same. Memorizing this approach helps you reliably deal with any octal number you encounter. This place-value calculation also parallels the same logic behind converting from binary to decimal or from hexadecimal to decimal, just with a different base.

Detailed Step-by-Step Conversion with Multiple Examples

The more you practice, the faster you’ll be at converting Octal to Decimal. Let’s walk through multiple examples of various complexities:

Example 1: Small Octal Number

• Octal Number: 7 (single digit)
• Interpretation: 7 × 8⁰ = 7 × 1 = 7 (decimal)

Yes, that’s almost trivial. Single-digit octal numbers are identical to their decimal counterparts as long as they’re within 0 to 7.

Example 2: Two-Digit Octal Number

• Octal Number: 15 (octal)
• Identify digits: 1 and 5
• Place values:
  • 1 is in the 8¹ position
  • 5 is in the 8⁰ position
• Calculation:
  • (1 × 8¹) + (5 × 8⁰)
  • (1 × 8) + (5 × 1) = 8 + 5 = 13 (decimal)

Hence, 15 in octal is 13 in decimal.

Example 3: Three-Digit Octal Number

• Octal Number: 325 (octal)
• Digits and positions:
  • 3 (8²), 2 (8¹), 5 (8⁰)
• Calculate each:
  • 3 × 8² = 3 × 64 = 192
  • 2 × 8¹ = 2 × 8 = 16
  • 5 × 8⁰ = 5 × 1 = 5
• Sum: 192 + 16 + 5 = 213 (decimal)

Example 4: Four-Digit Octal Number

• Octal Number: 0472 (octal)
• Notice the leading zero. This is common in many programming languages to signify an octal literal. Despite the prefix, treat it as 472 (octal).
• Positions:
  • 0 (8³), 4 (8²), 7 (8¹), 2 (8⁰)
• Calculations:
  • 0 × 8³ = 0
  • 4 × 8² = 4 × 64 = 256
  • 7 × 8¹ = 7 × 8 = 56
  • 2 × 8⁰ = 2 × 1 = 2
• Summation: 256 + 56 + 2 = 314 (decimal)

Here, 0472 (octal) converts to 314 in decimal.

Example 5: Large Octal Number

• Octal Number: 763214
• Positions: from right to left
  • 4 (8⁰), 1 (8¹), 2 (8²), 3 (8³), 6 (8⁴), 7 (8⁵)
• Stepwise:
  • 7 × 8⁵ = 7 × 32768 = 229376
  • 6 × 8⁴ = 6 × 4096 = 24576
  • 3 × 8³ = 3 × 512 = 1536
  • 2 × 8² = 2 × 64 = 128
  • 1 × 8¹ = 1 × 8 = 8
  • 4 × 8⁰ = 4 × 1 = 4
• Final Sum: 229376 + 24576 + 1536 + 128 + 8 + 4 = 255628 (decimal)

As shown above, even large octal values reliably convert to decimal with the same repeated approach.

The Relevance of Leading Zeros in Octal Values

You may commonly see octal numbers prefixed by a zero (e.g., 0755 for Unix file permissions). This is partially a historical artifact of certain programming languages derived from C, where a leading zero indicated that the number following it was octal rather than decimal. This can be a cause of subtle bugs if you’re not wary of them.

• Example: If you mistakenly type 010 in a code snippet, you might believe you are representing decimal 10, but in many languages that interpret leading zeros as octal, you’re actually representing 8 in decimal terms.

Being mindful of these leading zeros can prevent confusion. When in doubt, explicitly note that your number is octal or convert it to decimal for clarity in the code, especially if your team or environment might misread your intention.

Common Use Cases in Modern Systems

1. Unix and Linux File Permissions
   When you run the command chmod 755 filename in Unix-like systems, you’re expressing these permissions in octal. 755 indicates that the owner has read, write, and execute permissions (7 in octal), while groups and others have read and execute permissions only (5 in octal). If you wanted to be absolutely precise in decimal, you could convert 7, 5, and 5 from Octal to Decimal, but practically we maintain the 7-5-5 format in its octal representation because file systems interpret it that way.

2. Logging and Memory Dumps
   Certain low-level debugging tools, especially older ones, might automatically display memory addresses or data in octal. If you need to cross-verify those addresses with decimal offsets, you’ll have to perform an Octal to Decimal conversion. That’s why many developers who work with legacy systems keep a mental note of how to convert quickly.

3. Firmware Updates in Embedded Devices
   For specialized microcontrollers or smaller embedded electronics, you might occasionally see firmware data in octal format, especially if the device or manufacturing environment dates back to the times when octal was more mainstream. The necessity to interpret these numbers in decimal can crop up when performing calculations or analyzing memory maps.

4. Educational Settings
   Classrooms and textbooks that aim to teach fundamental computing concepts find octal a natural stepping stone between binary (base-2) and hexadecimal (base-16). Understanding the synergy between base-2, base-8, and base-16 fosters a deeper comprehension of how computers handle data. Assignments will likely include the Octal to Decimal conversion process as a core exercise.

Methodical Approaches to Conversion

While the standard approach outlined (positional multiplication and summation) is the most direct, there are a few conceptual pathways you can leverage:

1. Direct Positional Notation (the approach we’ve hammered thoroughly).

2. Binary Intermediary: Because each octal digit neatly maps to exactly three binary digits, you can convert octal to binary first, then convert the resulting binary to decimal. Although it may seem indirect, in certain mental math or code-based approaches, it’s sometimes easier if you recall quick binary to decimal conversions. For instance:

   • For octal digit 7, the binary representation is 111.
   • For octal digit 3, the binary representation is 011. (Leading zeros can be relevant in the grouping of three bits.)
     From that binary form, you can then convert to decimal.

   This binary intermediary approach was particularly popular back when the direct correlation between octal digits and 3-bit groupings in machine code was crucial. But for everyday conversions, the direct positional method is simpler.

3. Leveraging Programming Languages: Modern high-level languages like Python, JavaScript, C#, etc., have built-in or easily codable methods to handle conversions from Octal to Decimal. For instance, in Python, you can treat an octal literal with a specific prefix (0o) or simply parse the string as base-8 using int("123", 8). In that environment, manual conversions are rarely necessary. However, understanding the mechanics behind it can still be helpful for debugging or specialized tasks.

Practical Examples in Various Programming Languages

In real-world scenarios, you might rely more often on your programming language’s standard libraries than manual arithmetic. Below are concise demonstrations in a few languages, purely for instructive purposes. Remember that each code snippet basically does the same job: interpreting an octal string in base-8 and outputting a decimal result.

Python

octal_string = "755"
decimal_value = int(octal_string, 8)
print(decimal_value)  # Outputs 493 (decimal)

The snippet above quickly converts the octal string "755" (often signifying file permissions) into its base-10 integer equivalent, 493.

JavaScript

let octalString = "0755";
let decimalValue = parseInt(octalString, 8);
console.log(decimalValue); // Outputs 493

JavaScript’s parseInt function, provided a second argument specifying base-8, interprets your string as an octal number.

C

#include <stdio.h>
#include <stdlib.h>

int main() {
    char octalString[] = "755";
    int decimalValue = strtol(octalString, NULL, 8);
    printf("%d\n", decimalValue); // 493
    return 0;
}

C’s strtol function can parse the string in base-8 to produce its decimal representation.

In each example, you’re effectively letting the language handle the detail of the Octal to Decimal conversion under the hood. That can be infinitely more convenient than reading octal numbers manually, especially when dealing with large datasets.

Avoiding Common Pitfalls

1. Misinterpretation of Leading Zeros: Many programming languages interpret numbers with leading zeros as octal. If you inadvertently place a zero in front of a decimal integer, you might trigger an unwanted Octal to Decimal conversion and cause cryptic bugs. This problem was common in older JavaScript versions and is partially rectified in more standardized implementations that require a specific prefix or the usage of parseInt(num, 10) for decimal.

2. Range Errors: Octal digits only range from 0 to 7. If you mistakenly insert an 8 or 9 in an octal sequence (e.g., 089), some parsers will either throw an error or interpret the digit differently. Always ensure your octal strings truly contain only [0-7] to avoid undefined behavior or unexpected conversions.

3. Overlooking Large Numbers: When dealing with extremely large octal values—possibly beyond standard integer ranges—use data types capable of handling big integers or bignums, depending on your language of choice. Standard 32-bit integer ranges may overflow if your octal input is large enough. Tools or languages that support arbitrary-precision integer arithmetic (like Python’s default int) can help circumvent these issues.

4. Forgetting About Endianness: Typically, endianness refers to how bytes are ordered in memory rather than how numbers themselves are displayed, but occasionally, misunderstandings about how data is stored or represented at a low level leads to confusion about how to interpret octal strings. Usually, keep the concept of endianness separate from textual representation. Rely on consistent processes for conversions to avoid mistakes.

Utilizing Online Tools and Calculators

If you’d rather not open a programming environment or do the math by hand, many websites offer free Octal to Decimal converters. They’ll take your octal input, parse it in real time, and return the decimal result. Such tools can be handy for quick checks or letting newcomers practice conversions. Some of these sites also offer features like:

• Converting decimal numbers to octal.
• Converting to and from hexadecimal or binary.
• Handling large numbers that exceed typical integer boundaries in certain languages.
• Providing step-by-step outlines of the conversion process to help you learn.

Using an online or offline calculator can expedite your workflow if you frequently handle conversions or if your project demands quick numeric translations. However, having a fundamental understanding of the process is always beneficial, so you know how to interpret the outcomes and can troubleshoot if anything goes awry.

Advanced Computation: Converting Fractional Octal to Decimal

Though not as common in day-to-day computing tasks, fractional octal numbers also exist and can be converted to decimal. For instance, octal numbers can appear in forms like 25.74, with digits on both sides of a decimal (or more properly “octal point”). The method for handling the fractional portion extends the same base principle to negative powers of 8.

1. Integer Part: Convert using the standard integer positional method.
2. Fractional Part: Each digit after the “octal point” is multiplied by 8 raised to a negative power, starting at -1 for the first digit, -2 for the second, and so on.

For example, the octal number 25.74 splits into integer part 25 and fractional part 0.74.

• Integer Part (25):

  • 2 × 8¹ + 5 × 8⁰ = 16 + 5 = 21 (decimal)

• Fractional Part (.74):

  • 7 × 8⁻¹ + 4 × 8⁻²

Calculating further:

• 7 × 8⁻¹ = 7 ÷ 8 = 0.875
• 4 × 8⁻² = 4 ÷ 64 = 0.0625

Sum of fractional part = 0.875 + 0.0625 = 0.9375

Adding the integer part (21) to the fraction (0.9375) yields 21.9375 in decimal. Although it’s not an everyday scenario for most people, it demonstrates how Octal to Decimal conversion extends beyond integers.

Bridging Binary, Octal, and Hexadecimal

Given that modern computing commonly uses hexadecimal to represent memory addresses, color codes in web design (#FFFFFF for white, for instance), and certain other features, you might wonder how octal still competes. One of the main takeaways from this entire discussion is that each base has its niche:

• Binary (Base-2): The raw language of computers.
• Octal (Base-8): Maps neatly to groupings of three bits. Historically used in certain architectures, file permissions in Unix, certain embedded systems, and educational settings.
• Decimal (Base-10): Widely used by humans in everyday mathematics.
• Hexadecimal (Base-16): Efficient for representing bytes and memory addresses, widely used in modern computing.

Thus, it’s beneficial to have universal knowledge about each system, especially if you plan to delve more deeply into programming, computing, electronics, or advanced mathematics. Converting from Octal to Decimal is one piece of that bigger puzzle, letting you navigate seamlessly between these numerical worlds.

Real-World Case: Octal in Microcontroller Configurations

To highlight the significance of Octal to Decimal conversions, consider a scenario involving a small microcontroller in a retro engineering project. For instance, you’re working with an older device that expects line configurations in octal form. Suppose you have some sensor addresses indicated as 0752, and you need to match that up with an internal table that shows addresses in decimal. You realize:

1. 0752 is octal, signifying:
   • 0 × 8³ = 0
   • 7 × 8² = 448
   • 5 × 8¹ = 40
   • 2 × 8⁰ = 2
   • Total = 490 (decimal)

If your sensor’s register map or device datasheet references decimal addresses, you’d enter 490. Without that conversion, you could be writing data to the wrong register, causing malfunctions. The octal prefix 0 is an immediate clue that the number is octal, but it might be overlooked if you’re unacquainted with that convention. This is precisely how mistakes occur in a real engineering context.

The Education Factor

From a pedagogical standpoint, working through base conversions is not purely an anachronistic exercise leftover from computing in the 1970s. It underpins big computer science topics, including how we interpret data in memory and how computations are organized at the machine code level. Labs and tutorials that walk students through binary, octal, decimal, and hexadecimal conversions impart a stronger understanding of how digital systems interpret the concept of “numbers.”

Moreover, the skill of quickly shifting between bases cultivates mental dexterity. It strengthens an appreciation for place value, fosters an understanding of computational logic at the bit level, and helps future translators of code, compilers, or interpreters. Even if direct usage might fade with more user-friendly systems, the knowledge endures as a valuable resource for problem-solving.

Debugging in Environments Still Dependent on Octal

Many professionals might assume that any modern environment would prefer hexadecimal or decimal. But if you happen to maintain a large legacy codebase, you may find years (or even decades) worth of file I/O routines or system calls that lean on octal. Concomitantly, debugging logs might spit out octal references to memory offsets. If you’re working in such an environment, not comprehending how to convert Octal to Decimal can be a handicap. You’d either be forced to rely heavily on external tools or be stumped when quick mental checks are necessary. Hence, the skill has a direct effect on time-to-fix for debugging issues in systems that inherently rely on octal.

Large-Scale Data and Batch Conversions

Suppose your project or job requires you to process large sets of data, each containing numeric columns in octal form. If you need them in decimal, you might code or script a batch conversion pipeline (exploiting a language's native conversion function) or recursively call an online tool’s API if available. The practicality of a straightforward formula or function call cannot be understated in these contexts. Speed, accuracy, and reliability govern successful projects, especially if you’re expecting hundreds of thousands of lines of data.

Potential Confusion with “Octal” in Differing Fields

Within strictly mathematical contexts, “octal” is universally base-8. However, in broader engineering fields, sometimes “octal” can refer to connector pinouts, vacuum tube references, or other mechanical or hardware-based naming. Conversions in those contexts might not strictly refer to numeric systems, but the confusion can exist. Make sure you confirm that when an engineer says “octal” in a circuit design context, they truly mean base-8 numeric data or an “octal socket.” The lines may blur, but at the heart of computing, “octal” numerically always means base-8.

Practical Tips for Quick Manual Conversions

1. Memorize Powers of 8: Having quick recall for small powers of 8 (1, 8, 64, 512, 4096, 32768, etc.) helps.
2. Segmenting the Number: If you have a very large octal number, consider breaking it into smaller, more manageable chunks. For instance, convert the left half then shift to the right half, summing partial conversions at the end.
3. Cross-Verification: If you’re new to manual conversions, always cross-verify your result with a known tool or a smaller test.
4. Leverage the Binary Relationship: If you find binary to decimal conversions easier, break each octal digit into a group of three bits and proceed from there. For instance, the octal digit 5 is binary 101 and equals 1×2² + 0×2¹ + 1×2⁰ = 5 in decimal. This can be extended for each digit in an octal string.
5. Practice with Real Data: If you want to build confidence, find or create realistic practice scenarios. Maybe you attempt to decode file permissions from your own operating system, or you find an old engineering manual that references data in octal. Immersion cements skill.

Looking to the Future: Are Conversions Still Needed?

A natural question arises: “In an era where we have so many tools and such advanced hardware, do I really need to know how to convert from Octal to Decimal by hand?” The short answer is that, while it isn’t an everyday necessity for many developers, system administrators, or engineers, you might still encounter occasions where such knowledge is not just helpful but critical.

The dynamic and ever-evolving nature of technology means older systems remain in production or are repurposed. Meanwhile, knowledge gleaned from base conversions helps you understand deeper computing principles. If you’re building a broad skill set in computing, skipping base conversions is akin to skipping essential grammar lessons in foreign language study: you might get by with phrasebooks and guesswork, but you’d lack a crucial foundation that can hamper deeper understanding.

SEO Best Practices for an Octal to Decimal Page

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1. Keyword Placement: We have used the term “Octal to Decimal” multiple times in headings, subheadings, and body text. This consistent usage helps search engines recognize the central topic.
2. Synonyms and Related Terms: Words like “base-8,” “decimal system,” “number conversion,” and “file permissions” can reinforce the contextual relevance for search engines.
3. Use of Headings: Breaking the text into headings and subheadings makes it scannable for both humans and web crawlers.
4. Long-Form Content: Detailed articles generally rank better for comprehensive topics, catering to users who might want an in-depth explanation or quick reference.
5. User Engagement: Including real-world examples and code snippets keeps readers on the page longer if they’re actually looking for solutions. High dwell time can positively affect how search engines perceive content value.

By ensuring these best practices, your content stands a better chance of surfacing when users search for “Octal to Decimal” or related phrases.

Final Thoughts on Octal to Decimal

Converting Octal to Decimal may appear, at first glance, like an outdated skill. However, its pedagogical and technical value remains quite high in certain subdomains of computer science and information technology. Anyone working with legacy systems, microcontrollers, or Unix file permissions stands to benefit significantly from a thorough understanding of octal. On top of that, the knowledge fosters better overall comprehension of how different numerical bases operate, which is a strong asset in any computing or engineering role.

No matter your background or goals, it’s valuable to keep these takeaways in mind:

• The decimal system, though dominant, is just one of many representations humans use to interpret numbers.
• Octal has eight digits (0-7), linking it closely to binary groupings of three bits, which historically shaped its importance.
• The Octal to Decimal conversion process is straightforward and can be mastered quickly by methodically summing each octal digit multiplied by 8 raised to the digit’s positional power.
• Leading zeros or octal indicators in code can produce subtle bugs if misread, so always confirm your number’s base when working with certain programming languages.
• Real-world examples—in Unix permissions, embedded systems, and even educational labs—illustrate that octal remains relevant and beneficial in many contexts.

Whether your interest is purely academic, or you face an immediate practical need on the job, a deep understanding of how to convert octal numbers into decimal will make you a more versatile and confident practitioner. With each example and concept, the puzzle can become clearer—a single piece in the vast tapestry of computing knowledge. As you engage with different software, machines, or academic challenges, remembering the fundamentals of octal will prove time and again that no knowledge is truly obsolete, especially in a field that evolves but also respects its roots.

Shihab Ahmed

CEO / Co-Founder

Enjoy the little things in life. For one day, you may look back and realize they were the big things. Many of life's failures are people who did not realize how close they were to success when they gave up.