Addition
Overview
Addition is the process of combining two or more numbers to create a new value, and is generally considered the simplest form of mathematics. Despite its simplicity, the ability to perform basic addition is the foundation of most advanced mathematics, and simple addition, repeated millions of times per second, actually underlies much of the processing performed within the most advanced electronic computers on earth. Despite its elementary nature, the process of adding numbers together remains one of the most useful mathematical operations available, as well as perhaps the most common type of calculation performed on a daily basis by most adults.
Fundamental Mathematical Concepts and Terms
An addition equation requires only two terms to describe its component parts. When asked to name the simplest equation possible, most adults would respond with 1 + 1 = 2, probably the first math operation they learned. In this simple equation, the two 1s are termed addends, while the result of this or any other addition equation is known as the sum, in this case the value 2. Because this final value is called a sum, it is also correct, though less common, to describe the process of adding as summing, as in the expression, "Sum the five daily values to find the total attendance for the week." While the addition sign is properly called a plus sign, one does not ever refer to the process of addition as "plus-ing" two values.
A Brief History of Discovery and Development
Because the basic process of addition is so simple, its exact origins are impossible to identify. Near the beginning of recorded history, a variety of endeavors including commerce, warfare, and agriculture required the ability to add numbers; for some lines of work, addition was such a routine operation that specific tools became necessary in order to streamline the process. The most basic counting tools consisted of a small bag of stones or other small objects that could be used to tally an inventory of goods. In the case of shipping, a merchant counting sacks of grain as they were loaded onto his ship would move one small stone aside for each sack loaded, providing both a running total and a simple method to double-check the final count. Upon arrival, this same collection of stones would serve as the ship's manifest, allowing a
running count of the shipment as it was unloaded. In the case of warfare, a general might number his horses using this same method of having each object represented by a stone, a small seashell, or some other token. The key principle in this type of system was a one-to-one relationship between the items being counted and the smaller symbolic items used to maintain the tally.
Over time, these sets of counting stones gradually evolved into large counting tables, known as abaci, or in the singular form, an abacus. These tables often featured grooves or other placement aids designed to insure accuracy in the calculations being made, and tallies were made by placing markers in the proper locations to symbolize ones, tens, and hundreds. The counting tables developed in numerous cultures, and ancient examples survive from Japan, Greece, China, and the Roman Empire. Once these tables came into wide use, a natural evolution, much like that seen in modern computer systems, occurred, with the bulky, fixed tables gradually morphing into smaller, more portable devices. These smaller versions were actually the earliest precursors of today's personal calculator.
The earliest known example of what we today recognize as the hand-held abacus was invented in China approximately 5,000 years ago. Consisting of wood and moveable beads, this counting tool did not actually perform calculations, but instead assisted its human operator by keeping a running total of items added. The Chinese abacus was recognized as an exceptionally useful tool, and progressively spread throughout the world. Modern examples of the abacus are little changed from these ancient models, and are still used in some parts of the world, where an expert user can often solve lengthy addition problems as quickly as someone using an electronic calculator.
As technology advanced, users sought ways to add more quickly and more accurately. In 1642, a French mathematician Blaise Pascal (1623–1662) invented the first mechanical adding machine. This device, a complex contraption operated by gears and wheels, allowed the user to type in his equation using a series of keys, with the results of the calculation displayed in a row of windows. Pascal's invention was revolutionary, specifically because it could carry digits from one column to another. Mechanical calculators, the distant descendents of Pascal's design, remained popular well into the twentieth century; more advanced electrically operated versions were used well into the 1960s and 1970s, when they were replaced by electronic models and spreadsheet software.
In a strange case of history repeating itself, the introduction of the first high-priced electronic calculators in the 1970s was coincidentally accompanied by television commercials offering training in a seemingly revolutionary method of adding called Chisenbop. Chisenbop allowed one to use only his fingers to add long columns of numbers very quickly, and television shows of that era featured young experts out-performing calculator-wielding adults. Chisenbop uses a variety of finger combinations to represent different values, with the right hand tallying values from zero to nine, and the left hand handling values from ten and up. The rapid drop in calculator prices during this era, as well as the potential stigma associated with counting on one's fingers, probably led to the method's demise. Despite its seemingly revolutionary nature, this counting scheme is actually quite old, and may in fact predate the abacus, which functions in a similar manner by allowing the operator to tally values as they are added. Multiple online tutorials today teach the technique, which has gradually faded back into obscurity.
While the complex calculations performed by today's sophisticated computers might appear to lie far beyond anything achieved by Pascal's original adding machine, the remnants of Pascal's simple additions can still be found deep inside every microprocessor (as well as in a simple programming language which bears his name in honor of his pioneering work). Modern computers offer user-friendly graphic interfaces and require little or no math or programming knowledge on the part of the average user. But at the lowest functional level, even a cutting edge processor relies on simple operations performed in its arithmetic logic unit, or ALU. When this basic processing unit receives an instruction, that instruction has typically been broken down into a series of simple processes which are then completed one at a time. Ironically, though the ALU is the mathematical heart of a modern computer, a typical ALU performs only four functions, the same add, subtract, multiply, and divide found on the earliest electronic calculators of the 1970s. By performing these simple operations millions of times each second, and leveraging this power through modern operating systems and applications software, even a process as simple as addition can produce startling results.
Real-life Applications
SPORTS AND FITNESS ADDITION
Many aspects of popular sports require the use of addition. For example, some of the best-known records tracked in most sports are found by simply adding one success to another. Records for the most homeruns, the most 3-point shots made, the most touchdown passes completed, and the most major golf tournaments won in a career are nothing more than the result of lengthy addition problems stretched out over an entire career. On the business side of sports are other addition applications, including such routine tasks as calculating the number of fans at a ballgame or the number of hotdogs sold, both of which are found by simply adding one more person or sausage to the running total.
Many sports competitions are scored on the basis of elapsed time, which is found by simply adding fractions of a second to a total until the event ends, at which time the smallest total is determined to be the winning score. In the case of motor sports, racers compete for the chance to start the actual race near the front of the field, and these qualifying attempts are often separated by mere hundredths or even thousandths of a second. Track events such as the decathlon, which requires participants to attempt ten separate events including sprints, jumps, vaults, and throwing events over the course of two grueling days, are scored by adding the tallies from each separate event to determine a final score. In the same way, track team scores are found by adding the scores from each individual event, relay, and field event to determine a total score.
Although the sport of bowling is scored using only addition, this popular game has one of the more unusual scoring systems in modern sports. Bowlers compete in games consisting of ten frames, each of which includes up to two attempts to knock down all ten bowling pins. Depending on a bowler's performance in one frame, he may be able to add some shots twice, significantly raising his total score. For example, a bowler who knocks down all ten pins in a single roll is awarded a strike, worth ten plus the total of the next two balls bowled in the following frames, while a bowler who knocks down all ten pins in two rolls is scored a spare and receives ten plus the next one ball rolled. Without this scoring system, the maximum bowling score would be earned by bowling ten, ten-point strikes in a row for a perfect game total of 100. But with bowling's bonus scoring system, each of the ten frames is potentially worth thirty points to a bowler who bowls a strike followed by two more strikes, creating a maximum possible game score of 300.
While many programs exist to help people lose weight, none is more basic, or less liked, than the straightforward process of counting calories. Calorie counting is based on a simple, immutable principle of physics: if a human body consumes more calories than it burns, it will store the excess calories as fat, and will become heavier. For this reason, most weight loss plans address, at least to some degree, the number of calories being consumed. A calorie is a measure of energy, and 3,500 calories are required to produce one pound of body weight. Using simple addition, it becomes clear that eating an extra 500 calories per day will add up to 3,500 calories, or one pound gained, per week.
While this use of addition allows one to calculate the waistline impact of an additional dessert or several soft drinks, a similar process defines the amount of exercise required to lose this same amount of weight. For example, over the course of a week, a man might engage in a variety of physical activities, including an hour of vigorous tennis, an hour of slow jogging, one hour of swimming, and one hour officiating a basketball game. Each of these activities burns calories at a different rate. Using a chart of calorie burn rates, we determine that tennis burns 563 calories per hour, jogging burns 493 calories per hour, swimming burns 704 calories per hour, and officiating a basketball game burns 512. Adding these values up we find that the man has exercised enough to burn a total of 2,272 calories over the course of the week. Depending on how many calories he consumes, this may be adequate to maintain his weight. However if he is consuming an extra 3,500 calories per week, he will need to burn an additional 1,228 calories to avoid storing these extra calories as fat. Over the course of a year, this excess of 1,228 calories will eventually add up to a net gain of more than 63,000 calories, or a weight gain of more than 18 pounds.
While healthy activities help prolong life, the same result can be achieved by reducing unhealthy activities. Cigarette smoking is one of the more common behaviors believed to reduce life expectancy. While most smokers believe they would be healthier if they quit, and cigarette companies openly admit the dangers of their product, placing a health value or cost on a single cigarette can be difficult. A recent study published in the British Medical Journal tried to estimate the actual cost, in terms of reduced life expectancy, of each cigarette smoked. While this calculation is admittedly crude, the study concluded that each cigarette smoked reduces average life-span by eleven minutes, meaning that a smoker who puffs through all 20 cigarettes in a typical pack can simply add up the minutes to find that he has reduced his life expectancy by 220 minutes, or almost four hours. Simple addition also tells him that his pack-a-day habit is costing him 110 hours of life for each month he continues, or about four and one-half days of life lost for each month of smoking. When added up over a lifetime, the study concluded that smokers typically die more than six years earlier than non-smokers, a result of adding up the seemingly small effects of each individual cigarette.
FINANCIAL ADDITION
One of the more common uses of addition is in the popular pastime of shopping. Most adults understand that the price listed on an item's price-tag is not always the full amount they will pay. For example, most states charge sales tax, meaning that a shopper with $20.00 to spend will need to add some set percentage to his item total in order to be sure he stays under budget and doesn't come up short at the checkout counter. Many people estimate this add-on unconsciously, and in most cases, the amount added is relatively small.
In the case of buying a car, however, various add-ons can quickly raise the total bill, as well as the monthly payments. While paying 7% sales tax on a $3.00 purchase adds only twenty-one cents to the total, paying this same flat rate on a $30,000 automobile adds $2,100 to the bill. In addition, a car purchased at a dealership will invariably include a lengthy list of additional items such as documentation fees, title fees, and delivery charges, which must be added to the sticker price to determine the actual cost to the buyer.
As of 1999, Americans spent almost 40 cents of every food dollar at the 300,000 fast food restaurants in the country. Because they are often in a hurry to order, many customers choose one of the so-called value meals offered at most outlets. But in some cases, simple addition demonstrates that the actual savings gained by ordering a value meal is only a few cents. By adding the separate costs of the individual items in the meal, the customer can compare this total to learn just how much he is saving. He can also use this simple addition to make other choices, such as substituting a smaller order of French fries for the enormous order usually included or choosing a small soda or water in place of a large drink. Because most customers order habitually, few actually know the value of what they are receiving in their value meals, and many could save money by buying à la carte (piece by piece).
Deciding whether to fly or to drive is often based on cost, such as when a family of six elects to drive to their vacation destination rather than purchasing six airline tickets. In other cases, such as when a couple in Los Angeles visits relatives in Connecticut over spring break, the choice is motivated by sheer distance. But in some situations, the question is less clear, and some simple addition may reveal that the seemingly obvious choice is not actually superior. Consider a student living in rural Oklahoma who wishes to visit his family in St. Louis. This student knows from experience that driving home will take him eight hours, so he is enthusiastic about cutting that time significantly by flying. But as he begins adding up the individual parts of the travel equation, he realizes the difference is not as large as he initially thought. The actual flight time from Tulsa to St. Louis is just over one hour, but the only flight with seats available stops in Kansas City, where he will have to layover for two hours, making his total trip time from Tulsa to St. Louis more than three hours. Added to this travel time is the one hour trip from his home to the Tulsa airport, the one hour early he is required to check in, the half hour he will spend in St. Louis collecting his baggage and walking to the car, and the hour he will spend driving in St. Louis traffic to his family's home. Assuming no weather delays occur and all his flight arrive on time, the student can expect to spend close to seven hours on his trip, a net savings of one hour over his expected driving time. Simple addition can help this student decide whether the price of the plane ticket is worth the one hour of time saved.
In the still-developing world of online commerce, many web pages use an ancient method of gauging popularity: counting attendance. At the bottom of many web pages is a web counter, sometimes informing the visitor, "You are guest number …". While computer gurus still hotly debate the accuracy of such counts, they are a common feature on websites, providing a simple assessment of how many guests visited a particular site.
In some cases, simple addition is used to make a political point. Because the United States government finances much of its operations using borrowed money, concerns are frequently raised about the rapidly rising level of the national debt. In 1989, New York businessman Seymour Durst decided to draw attention to the spiraling level of public debt by erecting a National Debt Clock one block from Times Square. This illuminated billboard provided a continuously updated total of the national debt, as well as a sub-heading detailing each family's individual share of the total. During most of the clock's lifetime, the national debt climbed so quickly that the last digits on the counter were simply a blur. The clock ran continuously from 1989 until the year 2000, when federal budget surpluses began to reduce the $5.5 trillion debt, and the clock was turned off. But two years later, with federal borrowing on the rise once again, Durst's son restarted the clock, which displayed a national debt of over $6 trillion. By early 2005, the national debt was approaching $8 trillion.
POKER, PROBABILITY, AND OTHER USES OF ADDITION
While predicting the future remains difficult even for professionals such as economists and meteorologists, addition provides a method to make educated guesses about which events are more or less likely to occur. Probability is the process of determining how likely an event is to transpire, given the total number of possible outcomes. A simple illustration involves the roll of a single die; the probability of rolling the value three is found by adding up all the possible outcomes, which in this case would be 1, 2, 3, 4, 5, or 6 for a total of six possible outcomes. By adding up all the possibilities, we are able to determine that the chance of rolling a three is one chance in six, meaning that over many rolls of the die, the value three would come up about 1/6 of the time. While this type of calculation is hardly useful for a process with only six possible outcomes, more complex systems lend themselves well to probabilistic analysis. Poker is a card game with an almost infinite number of variations in rules and procedures. But whichever set of rules is in play, the basic objective is simple: to take and discard cards such that a superior hand is created. Probability theory provides several insights into how poker strategy can be applied.
Consider a poker player who has three Jacks and is still to be dealt her final card. What chance does she have of receiving the last Jack? Probability theory will first add up the total number of cards still in the dealer's stack, which for this example is 40. Assuming the final Jack has not been dealt to another player and is actually in the stack, her chance of being dealt the card she wants is 1 in 40. Other situations require more complex calculations, but are based on the same process. For example, a player with two pair might wonder what his chance is of drawing a card to match either pair, producing a hand known as a full house. Since a card matching either pair would produce the full house, and since there are four cards in the stack which would produce this outcome, the odds of drawing one of the needed cards is now better than in the previous example. Once again assuming that 40 cards remain in the dealer's stack and that the four possible cards are all still available to be dealt, the odds now improve to 4 in 40, or 1 in 10. Experienced poker players have a solid grasp of the likelihood of completing any given hand, allowing them to wager accordingly.
Probability theory is frequently used to answer questions regarding death, specifically how likely one is to die due to a specific cause. Numerous studies have examined how and why humans die, with sometimes surprising findings. One study, published by the National Safety Council, compiled data collected by the National Center for Health Statistics and the U.S. Census Bureau to predict how likely an American is to die from one of several specific causes including accidents or injury. These statistics from 2001 offer some insight into how Americans sometimes die, as well as some reassurance regarding unlikely methods of meeting one's end.
Not surprisingly, many people die each year in transportation-related accidents, but some methods of transportation are much safer than others. For example, the lifetime odds of dying in an automobile accident are 1 in 247, while the odds of dying in a bus are far lower, around 1 in 99,000. In comparison, other types of accidents are actually far less likely; for instance, the odds of being killed in a fireworks-related accident are only 1 in 615,000, and the odds of dying due to dog bites is 1 in 147,000. Some types of accidents seem unlikely, but are actually far more probable than these. For example, more than 300 people die each year by drowning in the bathtub, making the lifetimes odds of this seemingly unlikely demise a surprising 1 in 11,000. Yet the odds of choking to death on something other than food are higher by a factor of ten, at 1 in 1,200, and about the same as the odds of dying in a structural fire (1 in 1,400) or being poisoned (1 in 1,300). Unfortunately, these odds are roughly equivalent to the lifetime chance of dying due to medical or surgical errors or complications, which is calculated at 1 in 1,200.
USING ADDITION TO PREDICT AND ENTERTAIN
Addition can be used to predict future events and outcomes, though in many cases the results are less accurate than one might hope. For example, many children wonder how tall they will eventually become. Although numerous factors such as nutrition and environment impact a person's adult height, a reasonable prediction is that a boy will grow to a height similar to that of his father, while a girl will approach the height of her mother. One formula which is sometimes used to predict adult height consists of the following: for men, add the father's height, the mother's height, and 5, then divide the sum by 2. For women, the formula is (father's height + mother's height − 5) / 2. In most cases, this formula will give the expected adult height within a few inches.
One peculiar application of addition involves taking a value and adding that value to itself, then repeating this operation with the result, and so forth. This process, which doubles the total at each step, is called a geometric progression, and beginning with a value of 1 would appear as 1, 2, 4, 8, 16, 32 and so forth. Geometric progressions are unusual in that they increase very slowly at first, then more rapidly until in many cases, the system involved simply collapses under the weight of the total.
One peculiarity of a geometric progression is that at any point in the sequence, the most recent value is greater than the sum of all previous values; in the case of the simple progression 1, 2, 4, 8, 16, 32, 64, addition demonstrates that all the values through 32, when added, total only 63, a pattern which continues throughout the series. One seemingly useful application of this principle involves gambling games such as roulette. According to legend, an eighteenth century gambler devised a system for casino play which used a geometric progression. Recognizing that he could theoretically cover all his previous losses in a single play by doubling his next bet, he bragged widely to his friends about his method before setting out to fleece a casino. The gambler's system, known today as the Martingale, was theoretically perfect, assuming that he had adequate funds to continue doubling his bets indefinitely. But because the amount required to stay in the game climbs so rapidly, the gambler quickly found himself out of funds and deep in debt. While the story ends badly, the system is mathematically workable, assuming a gambler has enough resources to continue doubling his wagers. To prevent this, casinos today enforce table limits, which restrict the maximum amount of a bet at any given table.
Addition also allows one to interpret the cryptic-looking string of characters often seen at the end of series of motion picture credits, typically something like "Copyright MCMXXLI." While the modern Western numbering system is based on Arabic numerals (0–9), the Roman system used a completely different set of characters, as well as a different form of notation which requires addition in order to decipher a value. Roman numerals are written using only seven characters, listed here with their corresponding Arabic values: M (1,000), D (500), C (100), L (50), X (10), V (5), and I (1). Each of these values can be written alone or in combination, according to a set of specific rules. First, as long as characters are placed in descending order, they are simply added to find the total; examples include VI (5 + 1 = 6), MCL (1,000 + 100 + 50 = 1,150), and LIII (50 + 1 + 1 + 1 = 53). Second, no more than two of any symbol may appear consecutively, so values such as XXXX and MCCCCV would be incorrectly written.
Geometric Progression
An ancient story illustrates the power of a geometric progression. This story has been retold in numerous versions and as taking place in many different locales, but the general plot is always the same. A king wishes to reward a man, and the man asks for a seemingly insignificant sum: taking a standard chessboard, he asks the king to give him one grain of rice on day one, two grains of rice on day two, and so on for 64 days. The king hastily agrees, not realizing that in order to provide the amount of rice required he will eventually bankrupt himself.
How much rice did the king's reward require? Assuming he could actually reach the final square of the board, he would be required to provide 9,223,372,036,854,775,808 grains of rice, which by one calculation could be grown only by planting the entire surface of the planet Earth with rice four times over. However it is doubtful the king would have moved far past the middle section of the chessboard before realizing the folly of his generosity. The legend does not record whether the king was impressed or angered by this demonstration of mathematical wisdom.
Because these two rules are unable to produce certain values (such as 4 and 900), a third rule exists to handle these values: any symbol placed out of order in the descending sequence is not added, but is instead subtracted from the following value. In this way, the proper sequence for 4 may be written as IV (1 subtracted from 5), and the
Roman numeral for 900 is written CM (100 subtracted from 1,000). While this process works well for shorter numbers, it becomes tedious for longer values such as 1997, which is written MCMXCVII (1,000 − 100 + 1,000 − 10 + 100 + 5 + 1 + 1). Adding and multiplying Roman numerals can also become difficult, and most ancient Romans were skilled at using an abacus for this purpose. Other limitations of the system include its lack of notation for fractions and its inability to represent values larger than 1,000,000, which was signified by an M with a horizontal bar over the top. For these and other reasons, Roman numerals are used today largely for ornamental purposes, such as on decorative clocks and diplomas.
Potential Applications
While addition as a process remains unchanged from the method used by the ancient Chinese, the mathematical tools and applications related to it continue to evolve. In particular, the exponential growth of computing power will continue to radically alter a variety of processes. Gordon Moore, a pioneer in microprocessor design, is credited with the observation that the number of transistors on a processor generally doubles every two years; in practice, this advance means that computer processing power also doubles. Because this trend follows the principle of the geometric progression, with its doubling of size at each step, expanding computer power will create unexpected changes in many fields. As an example, encryption schemes, which may use a key consisting of 100 or more digits to encode and protect data, could potentially become easily decipherable as computer power increases. The rapid growth of computing power also holds the potential to produce currently unimaginable applications in the relatively near future. If the consistent geometric progression of Moore's law holds true computers one decade in the future will be fully 32 times as powerful as today's fastest machines.
