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Numeration Systems
19 questions By Tony R. Kuphaldt
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Question 7 of 19
What is the largest number that can be counted to, in a base-ten system with six digits? How about in a base-twenty (vigesimal) system with four places? How about in a base-two (binary) system with ten bits?
Reveal answer99999910 = Nine hundred ninety nine thousand, nine hundred ninety nine.
11111111112 = one thousand twenty three.
Notes:Ask your students to explain why subscripts are desirable when writing numbers in binary, or in any other “base” numeration system for that matter.
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Question 8 of 19
How many binary bits are needed to count up to the number one million three hundred thousand seven hundred sixty two? Try to answer this question without converting this quantity into binary form, and then explain the mathematical procedure!
Reveal answerTwenty one bits.
Hint: the answer consists in solving this equation:
1300762 = 2n Where,
n = The number of binary bits necessary to count up to 1,300,762.
Notes:This question is a test of whether or not students know how to algebraically solve for variable exponents. I’m not going to suggest how this is done (lest I rob a student of the learning experience that comes from researching the answer on their own), but I will say that it is a very useful algebraic technique, once mastered.
If students still haven’t found a solution after doing their research, suggest they try to solve a simpler problem:
- How many decimal digits are needed to count up to the number one million three hundred thousand seven hundred sixty two?
This question is trivial to answer (7 decimal digits), since we’re all familiar with decimal numeration. However, the real learning takes place when students write a mathematical expression for solving this problem, similar to the one written in the answer for the binary problem. Once they have that expression written, ask them what algebraic techniques could be used to solve for the exponent’s value.
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Question 9 of 19
The circuit shown in this diagram is used to transmit a numerical value from one location to another, by means of switches and lights:
Given the switches and lights shown, any whole number between 0 and 999 may be transmitted from the switch location to the light location.
In fact, the arrangement shown here is not too different from an obsolete design of electronic base-ten indicators known as Nixie tube displays, where each digit was represented by a neon-filled glass tube in which one of ten distinct electrodes (each in the shape of a digit, 0-9) could be energized, providing glowing numerals for a person to view.
However, the system shown in the above diagram is somewhat wasteful of wiring. If we were to use the same thirty-one conductor cable, we could represent a much broader range of numbers if each conductor represented a distinct binary bit, and we used binary rather than base-ten for the numeration system:
How many unique numbers are representable in this simple communications system? Also, what is the greatest individual number which may be sent from the “Sender” location to the “Receiver” location?
Reveal answerIn this system, with a “width” of 30 bits, we are able to represent one billion, seventy three million, seven hundred forty one thousand, eight hundred twenty four unique numbers. The greatest individual number which may be communicated is one less than this total.
Notes:The purpose of this question is to allow students to consider an electric circuit that communicates digital quantities from one location to another, rather than abstractly discuss numeration systems. It also provides a more practical context in which to understand maximum count in a numeration system.
