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Why We Should Switch to Hexadecimal

Why even broach the subject of changing something as ingrained as our standard system of numeration? The answer lies in one simple word: ergonomics. Merriam-Webster defines ergonomics as "an applied science concerned with designing and arranging things people use so that the people and things interact most efficiently and safely -- called also human engineering". Ergonomics as a principle has accompanied humankind since the dawn of invention. How does ergonomics relate to hexadecimal? Quite simply, hexadecimal optimizes the efficiency of using numbers. Greater efficiency means less wasted time, less work, less hassle, and more productivity. Let's examine the intrinsic human appeal of base sixteen.

Visual Perception

The human sense of visual perception possesses the extraordinary ability to divide by two. If you view a line segment, you can visually determine its midpoint with remarkable accuracy. Humans have a certain subconscious respect for binary symmetry, and this manifests itself in our skill of visual halving.

Hexadecimal maximizes the advantage of our natural instinct for symmetry because of the inherent symmetry of the number sixteen itself. A quantity of sixteen can be divided perfectly in half, those halves can be divided perfectly in half, the resulting quarters can be divided perfectly in half, and so on down to the lowest whole division. The number ten is evenly divisible into two parts of five, but five cannot be evenly divided in two. Our visual sense is not tuned to readily perceive divisions of five. In other words, base sixteen takes advantage of the intrinsic properties of human visual perception, whereas ten does not.

To understand this concept better, try a simple experiment yourself. Take a line segment and divide it evenly into fifths without using a ruler. Take another line segment and divide it evenly in half without using a ruler. Which one is more accurate? If you are using a Java enabled browser, click on the image at right to try this experiment right now. When the applet loads, just use your mouse to divide each line segment into fractions, then press the "Score" button to calculate your error.

What is the practical significance of this? It is easy for a human to divide a length into even parts of sixteen without needing a ruler. That cannot be said for ten. It is also easier to visualize or demarcate fractions of sixteen between two marks on a ruler. That would give a ruler with units based on sixteen better resolution than a ruler with units based on ten. It is also easier to divide a circle into even segments of sixteen.

Information Content

Let's compare two numbers: 6AE9BC16 and 110101011101001101111002. Which is larger? The second number certainly looks larger. However, the suprising answer is that they both represent the exact same quantity expressed in two different bases: hex and binary, respectively. The binary representation is significantly longer than the hex one because binary numbers contain less information content per digit than hexadecimal numbers. A single digit in binary or base-two only represents one of two possible values. A single hex digit, however, represents one of sixteen possible values.

As a rule: the larger the radix, the higher the information content per digit, and hence the lower the number of digits required to represent a given quantity. Therefore, a quantity represented in base sixteen is numerographically shorter than its base-ten equivalent. Conversely, given a hex number and a decimal number with an equal number of digits, the hex number will contain more information content than the decimal one. The disparity in information content continues to grow exponentially with every additional digit.

Digital Technology

All modern digital technology, from computers to stereos, represents data in binary. Every letter in a text document on your hard drive and every sound sample in a song on your favorite CD are stored as nothing more than ones and zeros. Base two happens to be the radix of choice because it is the easiest way for modern paradigms in digital architecture to store, retrieve, and operate on digital information. Digital circuitry uses transistors which are electronically switched either on or off (1 or 0) to represent binary values. Magnetic storage media such as hard drives represent binary data with sectors that are magnetically polarized either positive or negative. Optical media such as CDs are thermally etched so that the various sectors are either reflective or unreflective.

How does binary relate to hexadecimal? Each digit of a hexadecimal number represents exactly four digits of a binary number. This property is due to the fact that 16 equals 24. Manual conversion between binary and hexadecimal is easy: all you have to do is substitute one hex digit for every four binary digits, or vice versa. See the figure below.

Binary Equivalent of Hex Digits

While binary is easy for computers, enormous strings of ones and zeros are a bit unwieldy for people to use. However, as we saw in the preceding section, hex numbers are significantly shorter than their binary equivalents. Because of the simple interchangeability of binary and hexadecimal, humans can read digital data in hex, which preserves the underlying binary format that computers use, while presenting the information in a more human-readable format.

Because of the vast importance of base two in modern technology, an understanding of binary and hexadecimal are considered elementary knowledge for electrical engineers, computer scientists, digital hardware technicians, and even most IT professionals. Occupations in these fields account for a large and growing number of jobs as society becomes increasingly dependant on advancing digital technology. Hexadecimal could even be a required skill for your next job.

The Finger Myth

"Base ten is best because I have ten fingers on which to count."

This presumed origin of our use of base ten still persists as an argument against changing to base sixteen. Ironically, base ten is not the most pragmatic way to count on your fingers. Let's look at two examples of other radices which provide a much more satisfying finger-counting experience.

If you count in base six, letting one hand represent the first numerical place (60) and the other hand represent the next (61), you can count up to 556 which is equivalent to thirty-five in base ten. Remember that the number of digits required for a given radix is equal to that radix. So, base six requires six digits (0 through 5), which can easily be represented by holding up the required number of fingers (0 through 5) on each hand. See the figure below.

Counting On Your Fingers In Base Six
One (16) Five (56) Six (106)
Ten (146) Twenty (326) Thirty-five (556)

For even more fun, you can opt to count on your fingers in binary. Just let each finger represent one place of a ten-digit binary number. In binary the only digits are ones and zeros, so if your finger is raised, let it represent a one; if it is not raised, let it represent a zero. In this fashion, you can actually count up to 1,02310 (29 + 28 + 27 + 26 + 25 + 24 + 23 + 22 + 21 + 20). However, this method of counting is not recommended, as the number four is prone to be misinterpreted by onlookers.

For more information on counting in other bases and to see an interactive demonstration, visit Intuitor's Binary Counting page.

Interestingly, the ancient Mayans did not limit themselves to fingers. They used a vigesimal (base twenty) system, apparently drawing inspiration from fingers and toes. Why then should we exclude other extremities? Clearly our finger-chauvinism is arbitrary.

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