The Decimal Numeral System:
The modern system of counting using precisely ten symbols appears to relate to the fact that human hands have ten fingers. Small children use their fingers to count, add, and subtract. In fact, this has been done since counting became part of human culture. The most accepted theory on Roman numerals is that they evolved from Etruscan symbols. However, a common sense folk etymology for Roman numerals attributes the Roman numeral “I” to the shape of a single finger and the Roman numeral “V” to a hand with five outstretched fingers. Two such hands united by crossed wrists make an “X”, the Roman numeral for ten. Contemporary systems of counting have gone far beyond Roman numerals and there is no finger on a hand that corresponds to the symbol “0.” However, the most prevalent numeral system still uses precisely ten symbols. This numeral system is called the Decimal Numeral System (deci being Latin for ten).
Having ten basic numbers or symbols, one of which represents nothing or zero, means that numbers greater than nine can be expressed only by combining symbols in some manner. The chosen way to count numbers after 9 is to use a second position to the left to represent numbers multiplied by ten. When ten is reached, the smallest non-zero symbol, “1”, is placed into this new position, letting the first position run again from 0 to 9. In this manner a representation for numbers 10 – 19 is obtained. When twenty is reached, the symbol to the left (the one multiplied by 10) changes to “2,” and the first position then runs from 0 to 9, representing the numbers 20 to 29, and so on. By 99, all possible combinations of symbols in two positions have been used, and a new position to the left is created in which the symbol occupying that place is multiplied by 100. Counting is continued by starting over again with the other two positions (101,102, 103, etc.). Every student knows how this works, but it is important to recall the underlying rationale in order to understand the meaning of exponents and the different numeral systems that will be discussed below.
The same procedure works in reverse for numbers smaller than one. The first place to the right of the decimal, includes symbols whose value is divided by ten, the second place to the right of the decimal is for symbols whose value is determined by dividing by 100, and so on. Thus, .1 means one-tenth, .01 means one one-hundredth, .101 means one-hundred and one, one thousandths or one tenth plus one one-thousandth.
How Exponents Work:
When very large or very small numbers are required, the place value system becomes difficult to use. Exponents, written as a superscript after the symbol, are a simpler way to express these numbers, e.g. 102. The exponent indicates how many times to multiply a number by itself. 102 is the same as 100, without any obvious advantage in writing it either way (three symbols are required in both cases). But take one million: 106 is definitely shorter to write than 1,000,000. Note how the exponent corresponds to the number of zeros to the right of the “1.” Numbers smaller than one can also be represented by powers of ten, using negative exponents, as a negative power means how often the number one is divided by itself: 10-2 equals 1/10 times 1/10, which is one divided twice by 10. This number can also be written as 1/100 or 0.01. Exponential notation is also an advantage for very small numbers: 10-6 is a short way to represent one millionth, or 0.000001. In the films both the outward and the inward journeys quickly reach scales which would be rather complicated to express in writing without using exponential notation of the “powers of ten.”
Any number of the decimal numeral system can be represented as a sum of powers of ten multiplied by the value of its digits:
1,354.95 = 1 x 103 + 3 x 102 + 5 x 101 + 4 x 100 + 9 x 10-1 + 5 x 10-2
Recall that any number to the power of 1 is equal to itself and any number to the power of 0 is equal to 1.
A number can be written several different ways using exponents of the base 10. For example, 200 can be written 2 x 102, 20 x 101 or 200 x 100 or even 2000 x 10-1. Scientists often must use very large or very small numbers. Scientific notation makes these numbers easier to use by requiring that the first number (called the coefficient or the significand) is always between 1 and 10, The second number (called the base) is expressed as a power of 10. Thus, scientific notation will be expressed as a x 10b with a always being a number between one and ten.
Here are some examples of scientific notation:
- The speed of light, a constant in the universe frequently employed in physics, is approximately 300,000,000 meters per second. In scientific notation, this number is written as 3 x 108. Note that there are eight zeroes in the number when it is written in place-value format.
- The mass of an electron is about 0.00000000000000000000000000000091093822 kg. In scientific notation, this is expressed as 9.1093822×10-31 kg. Note that the 9 has moved 31 places to the left (30 zeros before the decimal and then one zero after the decimal).
- The mass of the Earth is approximately 5,973,600,000,000,000,000,000,000 kg. The scientific notation for this number is 5.9736×1024 kg.
- The circumference of the Earth is about 40,000,000 m. In scientific notation: 4×107 m.
An alternate way to express numbers in scientific notation has arisen because superscripts are difficult for computers, typewriters and calculators. Thus, the letter E is sometimes substituted for the number “10.” Thus, the circumference of the Earth can be expressed as 4E7.
The process of converting numbers to scientific notation is simple. It is only a function of moving the decimal point to the position in which the coefficient will be between 1 and 10. The number of places that you must move to the left will be the power of ten which constitutes the exponent. For example, 153,000,000 will be given the scientific notation of 1.53 x 108. For numbers less than one, the movement is in the opposite direction.
Numeral Systems Based on 16 and 2:
With these rules one could imagine and construct an alternative system with either more or fewer symbols. One that is used in certain areas of mathematics and computing is the hexadecimal system, built upon sixteen symbols: the numbers 0 to 9 and the letters A to F. Using these symbols we can count to fifteen using single digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F, and it is only when we get to sixteen that we need a new position, where, as before, we place the number “1” followed by a “0”. Note that the number 10 in the hexadecimal system has the value of sixteen in the decimal system. In order to distinguish both notations, the hexadecimal “10” gets a subscript: 10hex.
In the hexadecimal system, any number can again be represented as a sum of powers of “sixteen,” multiplied by its symbols:
E,7B3 = E x 10hex3 + 7 x 10hex2 + B x 10hex1 + 3 x 10hex0
which is equivalent to:
14 x 163 + 7 x 162 + 11 x 161 + 3 x 160 = 59,315.
Hexadecimal “powers of ten” are actually powers of sixteen. See more examples explained with graphics at Hexadecimal Number System. Because higher values can be expressed with fewer digits than in the decimal system, the hexadecimal system is frequently used in programming languages for computers and internet communications, where the tightest packaging of information is advantageous. There are even iPhone applications that will display a clock in the hexadecimal system! See Hex Clock by Peter Elst.
Another interesting way to count is the binary numeral system, which uses only two symbols. It is the basis of any computing language because computers use tiny electrical currents and there are only two possible states of an electrical switch: on and off. The first computers were built with switches that were valves (open/closed). Later, electricity and electronic switches (transistors) were introduced. Quantum computers replace the concept of switches with that of quantum states of a particular property of electrons such as spin, but there are still only two possible states (up/down) which are represented with a binary numeral system. Binary encoded information is translated into decimal or hexadecimal formats, for display or transmission purposes once the computer has processed the information in the only format it can handle: binary.
The most common and practical choice of symbols for a binary system is “0” for “off” and “1” for “on.” With just two symbols we already need to introduce a new digit to represent number two! This means that “two” in the binary system requires the same shift to a second place to the left as the number 10 in the decimal system or the number 16 in the hexadecimal system. Binary numbers are identified using the subscript “2”: Thus, the number two in binary is expressed as 102. Any binary number can also be represented by exponential notation:
1101 = 1 x 1023 + 1 x 1022 + 0 x 1021 + 1 x 1020
Which is equivalent to:
1 x 23 + 1 x 22 + 1 x 20 = 8 + 4 + 0 + 1 = 13
Binary “powers of ten” are really powers of two.
There is no limit to the numeral systems one could devise. Our culture has settled on the decimal system, but there is another one that is still deeply rooted in our society since Roman times and before. There are 12 months in a year and two 12-hour periods in a day, eggs are sold by the dozen, there are 12 inches to a foot and 12 Pence in an old British Shilling. The choice of 12 is not random, as it is the smallest number that can be divided in halves, thirds and quarters, making it especially useful in trade and storage. The two extra symbols to complete the set of 12 are most commonly represented by A and B, but there are other alternatives. See more on the duodecimal system at Duodecimal System.
Historically, there are more complex systems, such as the Mayan which uses a 20-based numeral system (see The Maya Mathematical System) and the Babylonian base-60 system (see Babylonian numerals). If a Maya saw our base 10 numeral system, he’d ask us why we forgot to include our toes. If a Babylonian saw our base 10 numeral system, she might ask us how we got by with so few numbers.