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CHAPTER 2 Data Types and Variables
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is difficult to imagine digits beyond 9 Letters A through F are used for numbers 10 through 15 Take a look at the following example It probably resembles a strangely formatted algebra problem It isn t algebra because letters A and B are not placeholders for real digits These letters are real digits in the hex numbering system
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A + 1 B
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Let s convert the previous problem from the hexadecimal number system to the decimal numbering system so you can have a better understanding of the calculation Here s the same calculation in decimal:
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10 + 1 11
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The binary numbering system is probably the most talked about numbering system aside from the decimal numbering system because the binary numbering system is used to represent data and instructions inside a computer The binary numbering system consists of two digits: 0 and 1 When counting, you carry over 1 to the left column when you reach digit 1 and then begin counting again This is shown in the next example No doubt the next example looks strange, but that s because we tend to assume that all calculations are performed using the decimal numbering system:
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The binary numbering system is the natural choice for representing numbers inside a computer because a computer is really a bunch of tiny switches The binary digit 0 is used to represent a switch that is turned off, and the binary digit 1 represents a switch that is turned on Programmers call each of these switches a bit Those not familiar with computers usually think the term bit refers to a little bit of information That s incorrect The term bit is an abbreviation for binary digit It is important to understand that any number you can write using the decimal numbering system can also be written in other numbering systems, regardless of the number of digits used by a numbering system Likewise, any calculation that can be performed using the decimal numbering system can be performed using any numbering system This means you can perform complex calculus using the binary numbering system if you wish, although not many of us would do such a thing
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Playing with Numbering Systems
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You can show off your new understanding of number systems by using the Calculator that comes with Windows to convert from one numbering system to another Here s what you need to do: 1 Open the Windows Calculator You ll find it under Programs | Accessories 2 Select View from the menu bar and then Scientific from the drop-down menu The image of the calculator expands to show, among other things, the four numbers systems spoken about in this section 3 Select the Dec radio button, if it is not already selected, and enter the number 15 4 Select the Hex radio button, and the letter F appears, which is the equivalent of the decimal value 15 in hexadecimal 5 Select the Oct radio button, and the number 15 is converted to 17, which is how the number 15 is represented in the octal numbering system 6 Select the Bin radio button and the number 15 is converted to 1111 This is the binary numbering system s representation of the decimal value 15
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Computer memory is really a bunch of switches When you want to place a decimal number in memory, the number is converted to the binary numbering system s representation of the number and then switches (computer memory) are turned on and off to represent the number Therefore, if we wanted to store the number 15 in memory, the number is first converted to its binary equivalent, which is 1111 Next, four switches (computer memory) are turned on As you ll recall from the previous section, placing the switch in the on position represents a binary 1 Fortunately, you don t have to turn on or turn off any switches As you ll see later in this chapter, the Java compiler does this for you Computer memory is divided into groups of eight switches, called a byte Each group is uniquely identified inside the computer by an address called a memory address A byte is the minimum amount of switches (memory) used store a number This means that the decimal number 15 is represented as 00001111 in memory The first four zeros are called leading zeros Therefore, 1111 and 00001111 are equivalent The largest decimal value that can be stored in a byte of memory is 255, which is the equivalent of 11111111 in binary Numbers larger than 255 can be stored by using more than one byte of memory to store the number