Chapter 3 Binary System and Boolean Algebra

Chapter 3 Binary System and Boolean Algebra 3.1 The Decimal System Our present system of numbers has 10 separate symbols 0, 1, 2, 3,…, 9, which are called Arabic numerals. We would be forced to stop at 9 or to invent more symbols if it were not for the use of positional notation. An example of earlier types of notation can be found in Roman numeral, which are essentially additive: III=I+I+I, XXV=X+X+V. New symbols (X, C, M, etc.) were used as the numbers increased in value. Thus V rather than IIIII=5. The only importance of position in Roman numerals lies in whether a symbol precedes or follows another symbol (IV=4 and VI=6). 我们当前的数字系统有0、1、2、3….9十个单独的符号，称之为阿拉伯数字。如果不使用位置符号，我们数到9就被迫停下来，或发明更多的符号。在罗马数字里可以找到早期符号类型的例子，他们基本上是加法的：Ⅲ=Ⅰ+Ⅰ+Ⅰ，XXV=X+X+V。当数值增加时采用新符号（X、C、M等）。这样V就不是IIIII=５。罗马数字中位的唯一重要性在于这个符号处于另一个符号之前或之后（Ⅳ=4和Ⅵ=6）。 The clumsiness of this system can easily be seen if we try to multiply XII by XIV. Calculating with Roman numerals was so difficult that early mathematicians were forced to perform arithmetic operations almost entirely on abaci, or counting boards, translating their results back into Roman-number form. Pencil-and-paper computations are unbelievably intricate and difficult in such systems. In fact, the ability to perform such operations as addition and multiplication was considered a great accomplishment in earlier civilization. 如果你要用XIV乘XII ，很容易看出这个数字系统是笨拙的。用罗马数字计算太难了，以至于早期的数字家几乎完全被迫在算盘或演算板完成算术运算，然后再把结果翻译成罗马数字形式。在这样的数字系统中，纸和笔运算达到以难置信的复杂和困难程度。事实上，在早期文明中能进行这样的加法和乘法运算被看作是一项伟大的成就。 The great beauty and simplicity of our number system can now be seen. It is necessary to learn only the basic numerals and the positional notation system in order to count to any desired figure. After memorizing the addition and multiplication tables and learning a few simple rules, it is possible to perform all arithmetic operations. Notice the simplicity of multiplying 12×14 using the present system. 现在可以看到我们的数字系统的巨大优势和简单明了，为了要数到任意想到的数字，只需要学会基本数字和进位符号，再记住加法和乘法表及学会一些简单规则，就可能完成所有的算术运算。看一下用现在数制计算12×14的简单性。 The actual meaning of the number 168 can be seen more clearly if we notice that it is spoken as “one hundred and sixty-eight”. Basically, the number is a contraction of (1×100)+(6×10)+8. The important point is that the value of each digit is determined by its position. For example, the 2 in 2,000 has a different value than the 2 in 20. We show this verbally by saying “two thousand” and “twenty”. Different verbal representations have been invented for numbers from 10 to 20 (eleven, twelve), but from 20 upward we break only at powers of 10 (hundreds, thousands, millions, billions). Written numbers are always contracted, however, and only the basic 10 numerals are used regardless of the size of the integer written. The general rule for representing numbers in the decimal system using positional notation is as follows: 如果我们注意到说„一百六十八‟时，数字168的实际意义就能更清楚地看出来。基本上，这个数字是（1×100）+6×10）+8的紧缩形式。更重要的是每个数字的值由它的位置来决定。例如2000中的2和20中的２的值是不同的。我说„二千‟和„二十‟来口头表达这些。从10到20我们发明出不同的口头表示方式。但是从20往上起，我们只在10的权位上断开。书写出的数字总紧凑的，不论写出的整数大小，只用10个基本数字。十进制使用进位符号表示数字的通则是：

The integer digit in different position is expressed as an-l, an-2, …, a0 where “n” is the number of digits to the left of the decimal point.

不同位上的整数用an-l, an-2, …, a0表示，n表示十进制小数点左面数字的数量。 The base, or radix of a number system is defined as the number of different digits which can occur in each position in the number system. The decimal number system has a base, or radix, of 10. This means that the system has 10 different digits (0, 1, 2, …, 9), any one of which may be used in each position in a number. History records the use of several other number systems. The quinary system, which has 5 for its base, was prevalent among Eskimos and North American Indians. Examples of the duodecimal system (base 12) may be seen in clocks, inches and feet and in dozens or grosses.

基数是定义在数字系统中每一位上的不同数字。十进制数有一个10的基数，这表示它有１０个不同数字。（0、1、2….9），其中任意一个可以用在数字的每个位置上。历史上记录了使用过几种其它数制。五进制有5个数字作为基数，在爱斯基摩人和北美印第安人中流行，十二进制（12个基数）可以在钟表，英尺，英寸以及以打记数中看到。 3.2 The Binary System A seventeenth-century German mathematician Gottfried Wilhelm von Leibniz, was an advocate of the binary number system which has 2 for a base, using only the symbols 0 and 1. It seems strange for an eminent mathematician to advocate such a simple number system; it should be noted that he was also a philosopher. Leibniz‟s reasons for advocating the binary system seem to have been mystical. He felt there was great beauty in the analogy between zero, representing the void, and one, representing the Deity.