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Extensive books, articles and research have been done on this topic and could provide enough information to fill this entire course if we allowed it to.
The range and diversity of creative thought that has been used in the past to describe numbers and to count objects and people is staggering.
There is generally a lack of books and research material concerning the historical foundations of the Americas. The reasons for this may be twofold: first, it is thought that there was a lack of specialized mathematics in the American regions; second, many of the secrets of ancient mathematics in the Americas have been closely guarded.
Two researchers, Leland Locke and Erland Nordenskiold, have carried out research that has attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics.
These researchers have come to certain beliefs about the quipu that we will summarize here. It should be noted that the Incas did not have a complicated system of computation.
Where other peoples in the regions, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the simpler task of record-keeping.
We will describe them in more detail in a moment. However, they first often needed to do computations whose results would be recorded on quipu.
To do these computations, they would sometimes use a counting board constructed with a slab of stone. In the slab were cut rectangular and square compartments so that an octagonal eight-sided region was left in the middle.
Two opposite corner rectangles were raised. Another two sections were mounted on the original surface of the slab so that there were actually three levels available.
In the figure shown, the darkest shaded corner regions represent the highest, third level. The lighter shaded regions surrounding the corners are the second highest levels, while the clear white rectangles are the compartments cut into the stone slab.
Pebbles were used to keep accounts and their positions within the various levels and compartments gave totals. For example, a pebble in a smaller white compartment represented one unit.
Note that there are 12 such squares around the outer edge of the figure. If a pebble was put into one of the two white larger, rectangular compartments, its value was doubled.
When a pebble was put in the octagonal region in the middle of the slab, its value was tripled. If a pebble was placed on the second shaded level, its value was multiplied by six.
And finally, if a pebble was found on one of the two highest corner levels, its value was multiplied by twelve. Different objects could be counted at the same time by representing different objects by different colored pebbles.
Suppose you have the following counting board with two different kind of pebbles places as illustrated. Let the solid black pebble represent a dog and the striped pebble represent a cat.
How many dogs are being represented? Watch this short video lesson about Inca counting boards. You will find that this is a review of concepts presented here about counting boards.
This kind of board was good for doing quick computations, but it did not provide a good way to keep a permanent recording of quantities or computations.
For this purpose, they used the quipu. The quipu is a collection of cords with knots in them. These cords and knots are carefully arranged so that the position and type of cord or knot gives specific information on how to decipher the cord.
A quipu is made up of a main cord which has other cords branches tied to it. See pictures to the right. Locke called the branches H cords.
They are attached to the main cord. B cords, in turn, were attached to the H cords. Most of these cords would have knots on them. Rarely are knots found on the main cord, however, and tend to be mainly on the H and B cords.
Locke points out that there are three types of knots, each representing a different value, depending on the kind of knot used and its position on the cord.
The Incas, like us, had a decimal base-ten system, so each kind of knot had a specific decimal value. They would be on the upper levels of the H cords.
Sometimes long knots were used to represents tens and hundreds. Note that the long knot has several turns in it…the number of turns indicates which integer is being represented.
The units ones were placed closest to the bottom of the cord, then tens right above them, then the hundreds, and so on. In order to make reading these pictures easier, we will adopt a convention that is consistent.
For the long knot with turns in it representing the numbers 2 through 9 , we will use the following notation:.
The four horizontal bars represent four turns and the curved arc on the right links the four turns together. This would represent the number 4.
What numbers are represented on each of the four cords hanging from the main cord? The colors of the cords had meaning and could distinguish one object from another.
One color could represent llamas, while a different color might represent sheep, for example. When all the colors available were exhausted, they would have to be re-used.
Because of this, the ability to read the quipu became a complicated task and specially trained individuals did this job. They were called Quipucamayoc, which means keeper of the quipus.
They would build, guard, and decipher quipus. There were various purposes for the quipu. Some believe that they were used to keep an account of their traditions and history, using knots to record history rather than some other formal system of writing.
One writer has even suggested that the quipu replaced writing as it formed a role in the Incan postal system.
Yet another proposed use of the quipu was to record numbers related to magic and astronomy, although this is not a widely accepted interpretation.
The mysteries of the quipu have not been fully explored yet. We are so used to seeing the symbols 1, 2, 3, 4, etc. Unfortunately, as we proceed through our mathematical education in grade and high school, we receive very little information about the wide range of number systems that have existed and which still exist all over the world.
The fact that it has survived for hundreds of years and shows no sign of going away any time soon suggests that we may have finally found a system that works well and may not need further improvement, but only time will tell that whether or not that conjecture is valid or not.
We now turn to a brief historical look at how our current system developed over history. This is a base-ten decimal system since place values increase by powers of ten.
Furthermore, this system is positional, which means that the position of a symbol has bearing on the value of that symbol within the number.
For example, the position of the symbol 3 in the number , gives it a value much greater than the value of the symbol 8 in that same number. The development of these ten symbols and their use in a positional system comes to us primarily from India.
However, the history of these numbers and their development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni, whose picture is shown in figure When we look at the origins of the numbers that al-Biruni encountered, we have to go back to the third century BCE to explore their origins.
It is then that the Brahmi numerals were being used. The Brahmi numerals were more complicated than those used in our own modern system.
They had separate symbols for the numbers 1 through 9, as well as distinct symbols for 10, , ,…, also for 20, 30, 40,…, and others for , , , …, The Brahmi symbols for 1, 2, and 3 are shown below.
For example, in the first century CE, one particular set of Brahmi numerals took on the following form: . One of those paths led to our current numeral system, and went through what are called the Gupta numerals.
They have the following form: . How the numbers got to their Gupta form is open to considerable debate. Many possible hypotheses have been offered, most of which boil down to two basic types.
This is not uncommon. The second type of hypothesis states that they were derived from some earlier number system.
However, there are other hypotheses that are offered, one of which is by the researcher Ifrah. His theory is that there were originally nine numerals, each represented by a corresponding number of vertical lines.
One possibility is this: . Because these symbols would have taken a lot of time to write, they eventually evolved into cursive symbols that could be written more quickly.
If we compare these to the Gupta numerals above, we can try to see how that evolutionary process might have taken place, but our imagination would be just about all we would have to depend upon since we do not know exactly how the process unfolded.
The activity X of a sample of 99m Tc of unknown activity would then be given by. It is desirable to keep counting rates from background radiation as low as possible with the well counter to minimize statistical uncertainties in counting measurements see Chapter 9, Section D.
Sources of background include cosmic rays, natural radioactivity in the detector e. Additional sources of background in a hospital environment include patients who have been injected with radionuclides for nuclear medicine studies or for therapeutic purposes.
These sources of radiation, although usually located some distance from the counter, can produce significant and variable sources of background.
External sources of background radiation are minimized by surrounding the detector with lead. The thickness of the lead shielding is typically 2.
In well counters with automated multiple-sample changers Section A. In many nuclear medicine procedures, background counting rates are measured between samples, but if the background counting rate becomes large e.
Commercial well-counter systems have push-button or computer selection of the appropriate SCA window settings for different radionuclides. In these systems compensation has been made by the manufacturer for the nonlinear energy response of the NaI Tl detector.
However, because of the possibility of drifts in the electronics and the PM tube gain with time, the response of the well counter should be checked regularly with a long-lived standard source, such as Cs, as a quality assurance measure.
Some modern well counters incorporate MCAs, allowing the entire spectrum to be measured and analyzed. When multiple radionuclides are counted simultaneously e.
If SCA windows are positioned on the 99m Tc window 1 and 51 Cr window 2 photopeaks, a correction for the interference can be applied as follows: A sample containing only 51 Cr is counted and the ratio R 12 of counts in window 1 to counts in window 2 is determined.
Similarly, a sample containing only 99m Tc is counted and the ratio R 21 of counts in window 2 to counts in window 1 is determined.
Suppose then that a mixed sample containing unknown proportions of 99m Tc and 51 Cr is counted and that N 1 counts are recorded in the 99m Tc window window 1 and that N 2 counts are recorded in the 51 Cr window window 2.
Suppose further that room and instrument background counts are negligible or have been subtracted from N 1 and N 2.
Then the number of counts from 99m Tc in window 1 [ N 1 99m Tc ] can be calculated from. Crosstalk from 51 Cr into the 99m Tc window must be corrected for, using methods described in the text.
Equations and permit calculation of the number of counts that would be recorded in the photopeak window for each radionuclide in the absence of crosstalk interference from the other radionuclide.
These equations can be used for other combinations of radionuclides and window settings with appropriate changes in symbols. For greatest precision, the ratios R 12 and R 21 should be determined to a high degree of statistical precision e.
Generally, the technique is not reliable for the in vivo measurements described in Section F, because of varying amounts of crosstalk caused by Compton scattering within body tissue.
A mixed sample containing 99m Tc and 51 Cr provides 18, counts in the 99m Tc window and counts in the 51 Cr window. A sample containing 51 Cr alone gives 25, counts in the 51 Cr window and 15, crosstalk counts in the 99m Tc window, whereas a sample containing 99m Tc alone gives 20, counts in the 99m Tc window and crosstalk counts in the 51 Cr window.
What are the counts due to each radionuclide in their respective photopeak windows? Assume that background counts are negligible. The crosstalk interference factors are, for 51 Cr crosstalk in the 99m Tc window.
Therefore the counts in the 99m Tc window from 99m Tc in the mixed sample are Equation If higher levels of activity are employed, serious dead time problems can be encountered see Chapter 11, Section C.
Samples with high counting rates require short counting times and provide good statistical precision with little interference from normal background radiation.
If only a few samples must be counted, they can be counted quickly and conveniently using manual techniques; however, with long counting times or large numbers of samples, the counting procedures become time consuming and cumbersome.
Systems with automated sample changers have been developed to alleviate this problem Fig. Typically, these systems can accommodate or more samples, and each sample is loaded automatically into the counter in a sequential manner.
Hundreds of samples can be loaded and measured in a single run. This system also incorporates a multichannel analyzer for spectral analysis.
As shown in Figure , the sample hole passes through the entire length of the NaI Tl crystal, and the PM tube is connected to the side of the scintillator.
A key advantage of the through-hole detector is that samples can be automatically positioned at the center of the NaI Tl crystal, irrespective of sample volume.
This results in the highest detection efficiency and minimizes efficiency changes with volume. Figure shows the smaller changes in efficiency with volume for a through-hole versus a well-type counter for 59 Fe.
Placement of the sample can be automatically adjusted to center the sample volume in the detector. The efficiency of the through-hole detector shows less variation with sample volume because the sample can be centered in the detector.
Adapted from Guide to Modern Gamma Counting. Systems with automated sample changers not only save time but also allow samples to be counted repeatedly to detect variations caused by malfunction of the detector or electronic equipment or changes in background counting rates.
Background counting rates can be recorded automatically by alternating sample and blank counting vials.
In these systems, counting vials loaded into a tray or carriage are selected automatically and placed sequentially in the NaI Tl well counter.
Measurements are taken for a preset time or a preset number of counts selected by the user.