| The expression xn is called a power where, n is the exponent and x is the base. |
Example: 210= 1024, 1024 is a power of 2, specifically it is 2 multiplied by itself 10 times: 2•2•2•2•2•2•2•2•2•2. Exponentiation is a shorthand notation for such repeated multiplication.
Most people have 5 digits ("fingers") on each hand and 2 hands. This has led to the use of the decimal system of notation with 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We express our numbers using place value where each position to the left is weighted 10 times the position to its right. Thus 1331= 1•103 + 3•102 + 3•101 + 1•100 = 1000 + 300 + 30 + 1. This system of writing numbers is the Hindu-Arabic Number System or Arabic Numerals.
| "Anything" to the zero power is 1: x0 = 1 (x cannot equal 0). |
| Anything to the first power is itself: x1 = x. |
Properties of Exponents:
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Notice how the place value system was not possible before zero was invented/discovered! You should be familiar with the Roman Numeral System where I=1; V=5; X=10; L=50; C=100; D=500; and M=1000. Smaller values go to the right unless they represent subtraction. The restrictions for subtraction are: 1) you can subtract no more than one symbol; 2) that symbol can not be more than an order of magnitude less; and 3) it must also be a power of ten. Thus 49 = XLIX, but not IL.
| One order of magnitude means one power of ten. |
A Kei0 term is order of bagnitude,
or binary order of magnitude, which means one power of two.
The number above (1331) could just as easily be expressed in base 11 as 100011 = 1•113 + 0•112 + 0•111 + 0•110. Note: when no base is indicated (usually via a subscript afterwards), base 10 is assumed.
Maybe you prefer base 12, where 92E12 = 9•122 + 2•121 + E•120, and T represents the digit "ten" and E represents "eleven" in our duodecimal system. The following example also illustrates how to convert from base 10 to another base by repeated division and use of the remainders.
| _ | _ | 9 | R | 2 | ||||||
| 1 | 2/ | 1 | 1 | 0 | R | 1 | 1 | (or "E") | ||
| 1 | 2 | /1 | 3 | 3 | 1 |
The computer revolution has expanded the use of bases 2, 8, and 16 especially. A typical base 2 number might be (the character "6" in EBCDIC):
| 111101102 | = | 1•27 + 1•26 + 1•25 + 1•24 + 0•23 + 1•22 + 1•21 + 0•20 |
| = | 128 + 64 + 32 + 16 + 0 + 4 + 2 + 0 | |
| = | 246 |
Base 2 is also called binary. Base 8 is known as octal. Hexadecimal, or affectionately called hex for short, refers to base 16. Since 4, 8, and 16 are powers of 2, it is an easy matter to convert such a number from base 2 to base 2n. You regroup bits n at a time from the right. For example: 111101102 = 33124 = 3668= F616. In base 16, we need names for our 6 additional "fingers" (I mean digits). The usual choices are A, B, C, D, E, and F. Below is a table of how the numbers are represented in the common bases.
| Each binary digit is called a bit. |
|
Each hexadecimal digit (or 4 bits) Kei It is more commonly called a nibble. |
| 8 bits make a modern byte. (Hence the term nibble above for half a byte.) |
| Note how close in magnitude 103= 1000 and 210= 1024 are. |
|
The term
kilo which really is 103 now often means 210 (1024). The term mega which really is 106 now often means 220 (1,048,576). The term giga which really is 109 now often means 230 (1,073,741,824). The term tera which really is 1012 now often means 240 (1,099,511,627,776?). |
| Base 16 | Base 10 | Base 2 |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
Parity is a term now commonly used in computer storage and communications. The word is related to par as in golf where "he hit under par" and connotes equivalence. In computers, it relates to base 2 and there are several types: even, odd, mark, and no. Even parity typically means a bit will be appended to each byte (or word) to force an even number of bits. For example, the character "1" in the ASCII communication code is 3116 or 001100012. If transmitted or stored with even partiy, this byte would have an additional bit appended and that bit would be set (=1) for there to be an even number of bits set. Odd parity would mean the appended bit would be reset (=0). Errors can then be detected if the received or recalled value does not have the correct parity. More advanced encoding schemes (Hamming, etc.) allow error correction as well, but require additional storage. Mark indicated the parity bit is always set (=1). No parity indicates the parity bit is either not present or equal to zero.
An interesting application of base 3, known as ternary, can be read about in an article in the American Scientist, July-Aug. 1998, pg 314-9. There is no reason the base has to be positive. A homework problem will deal with base -3. Base 60 was developed by the ancient Babylonians. We still use it for time (60 seconds = 1 minute; 60 minutes = 1 hour) and angle (60 seconds = 1 minute, 60 minutes = 1 degree; 6 •60 = 360 degree = 1 circle) measurements. A fun base can be base 26 and will also be dealt with in the homework. The letters of the English alphabet are an obvious choice for "digits".
Pierre de Fermat was an amateur mathematician living in the early 1600's who had a profound influence on mathematics for the last four centuries. Fermat noted that 220 + 1 = 3 = F0 was prime as was 221 + 1 = 5 = F1, 222 + 1 = 17 = F2, 223 + 1 = 257 = F3, 224 + 1 = 65537 = F4... He conjectured that 22n + 1 = Fn was always prime. In 1732, Leonard Euler, another famous mathematician, showed that 225 + 1 = F5 = 4294967297 was divisible by 641. The search for prime factors of larger Fermat Numbers continues and is another potential EXPO Project. In 1796, Gauss used Fermat numbers in his proof that a regular heptagon (7-sided polygon) was not constructible, whereas the regular heptadecagon (17-sided polygon) was. Please note that Fn usually refers to Fermat numbers and not Fibonacci numbers as presented in numbers lesson 2. (Note also: Most calculators process stacked exponents left to right and not right to left as mathematicians would expect, thus parentheses are highly recommended.)
Blaise Pascal was yet another famous mathematician contemporary with Fermat with whom he shares the honor of inventing probability. Pascal's triangle is useful in many diverse fields of mathematics and is displayed below:
| 1 | row 0 | ||||||||||||
| 1 | 1 | row 1 | |||||||||||
| 1 | 2 | 1 | row 2 | ||||||||||
| 1 | 3 | 3 | 1 | row 3 | |||||||||
| 1 | 4 | 6 | 4 | 1 | row 4 | ||||||||
| 1 | 5 | 10 | 10 | 5 | 1 | row 5 | |||||||
| .. | .. | .. | .. | .. | .. | .. |
Notice how each entry is the sum of the numbers diagonally above it to the left
and to the right—where missing numbers on the sides can be assumed to be
zero. Notice how we already saw the first few rows in the homework as the powers
of 11! Each entry in Pascal's triangle can also be found as:
| nCr | = | n! |
| r!(n-r)! |
Although we will formally define binomial in lesson 12, a quick review of algebra will be included here. First, when dealing with equations, it is important to always follow the "golden rule": "what you do to one side, always do to the other." This is partially formalized as two axioms as follows:
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Additive Property of Equality: If a = b, then a + c = b + c. Multipicative Property of Equality: If a = b, then ac = bc. |
Also, notice what happens when 23 is multiplied by 12:
| (12)(23) | = (10•20) + (10•3) + (2•20) + (2•3) | or another method | 23 | or | 12 |
| = 200 + 30 + 40 + 6 | ×12 | ×23 | |||
| = 276 | 46 | 36 | |||
| +23 | +24 | ||||
| 276 | 276 |
This is an important algorithm to remember when multiplying binomials such as (x + 1)(x + 1) = x2 + x + x + 1 = x2 + 2x + 1, and is often referred to as the FOIL method, an acronym for First, Outer, Inner, Last. However, the box method generalizes to higher order polynomials.
| 2x | -3y | |
| x | 2x2 | -3xy |
| -2y | -4xy | 6y2 |
So, (2x-3y)(x-2y)=2x2 - 7xy + 6y2
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