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Homeschool Fast Facts! |
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| For those of you just itching to convert some numbers, we offer the following calculator. |
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| Intro | Decimal | Binary | Octal/Hex | Powerful 2 | Storage Facts |
| Introduction to Different Numbering Systems |
| The Binary System is based on two numbers (base 2) and you probably know that digital computers use the binary system. Digital computers contain hundreds of thousands of transistors that are really just tiny "on" and "off" switches. Digital computers are designed to work with information as patterns of "on" and "off" signals which are represented in binary as 1's and 0's. People in the computer industry also use two other numbering systems as sort of a short-hand for the binary numbers. These systems are the Octal System (base 8) and the Hexadecimal System (base 16). We will cover these below. |
| What You Already Know |
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You already know something about the binary system because its rules are just like the rules for the decimal system. So let's first review the decimal system.
When you look at a number like 2001, you automatically know that it means "two thousand and one" things. The things may be years, pennies, apples, it doesn't matter. You know the size of a number based on the number of digits and you interpret the value based on the position of each digit. The positions increase in value from right to left as demonstrated in the tables below. |
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| The Decimal System | |||||||||
| The Decimal has a base of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, a number like "0123456789" has ten positions and each position can contain the digits 0-9. Although you know what the decimal number is by experience, you need to know how the decimal number is calculated in order to move on to the Binary System. | |||||||||
| digit value = digit * 10x where x = (position number - 1). It sounds complicated but look at the table below. | |||||||||
| Position | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
| Power of 10 | 108 | 107 | 106 | 105 | 104 | 103 | 102 | 101 | 100 |
| Decimal Value of a 1 in this position | 100,000,000 | 10,000,000 | 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
| Tip: any number raised to an exponent of zero = 1 (examples: 160 = 1; 100 = 1; 80 = 1; 20 = 1) | |||||||||
| Decimal Number Examples | |
| decimal number | = value by position (digit * value) |
| 11 | = (1 * 10) + (1 * 1) |
| 404 | = (4 * 100) + (0 * 10) + (4 * 1) |
| 2001 | = (2 * 1000) + (0 * 100) + (0 * 10) + (1 * 1) |
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| The Binary System | |||||||||
| The Binary System has a base of two numbers: 1 and 0. For example, a number like "1010" has four positions and each position can contain the digits 0-1. Most of us humans have a tough time relating to binary numbers. It helps to better understand them if you know how they convert them to a decimal number. | |||||||||
| digit value = digit * 2x where x = (position number - 1). See, it works just like the Decimal System only the base number is 2 rather than 10. | |||||||||
| Position | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | |
| Power of 2 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | |
| Decimal Value of a 1 in this position | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |
| Tip: any number raised to an exponent of zero = 1 (examples: 160 = 1; 100 = 1; 80 = 1; 20 = 1) | |||||||||
| Converting Binary Numbers to Decimal | ||
| binary number | decimal value by position (digit * value) | = decimal value |
| 0001 | (0 * 8) + (0 * 4) + (0 * 2) + (1 * 1) | 1 |
| 1010 | (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1) | 10 |
| 1111 | (1 * 8) + (1 * 4) + (1 * 2) + (1 * 1) | 15 |
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| The Octal and Hexadecimal Systems | |||
| The Octal (base 8) and Hexadecimal (base 16) System work just like the decimal and binary system. The Octal System uses 8 digits: 0, 1, 2, 3, 4, 5, 6, and 7. The Hexadecimal uses 16 digits but that poses a bit of a problem - there are more digits than available numbers. To solve the problem, the hexadecimal system uses a combination of 10 numbers (0-9) and 6 letters (A-F) to represent the 16 digits (see table below). |
| Number System Equivalents. | ||||||||
| Decimal | Binary | Octal | Hexadecimal | |||||
| 0 | 0000 | 0 | 0 | |||||
| 1 | 0001 | 1 | 1 | |||||
| 2 | 0010 | 2 | 2 | |||||
| 3 | 0011 | 3 | 3 | |||||
| 4 | 0100 | 4 | 4 | |||||
| 5 | 0101 | 5 | 5 | |||||
| 6 | 0110 | 6 | 6 | |||||
| 7 | 0111 | 7 | 7 | |||||
| 8 | 1000 | 10 | 8 | |||||
| 9 | 1001 | 11 | 9 | |||||
| 10 | 1010 | 12 | A | |||||
| 11 | 1011 | 13 | B | |||||
| 12 | 1100 | 14 | C | |||||
| 13 | 1101 | 15 | D | |||||
| 14 | 1110 | 16 | E | |||||
| 15 | 1111 | 17 | F | |||||
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| Binary Power | ||
| Bytes | Bits | |
| 1 bit | 0.125 byte | 1 bit |
| 1 byte | 1 byte | 8 bits |
| 1 kilobyte | 1 024 bytes | 8 192 bits |
| 1 megabyte | 1 048 576 bytes | 8 388 608 bits |
| 1 gigabyte | 1 073 741 824 bytes | 8 589 934 592 bits |
| 1 terabyte | 1 099 511 627 776 bytes | 8 796 093 022 208 bits |
| 1 petabyte | 1 125 899 906 842 624 bytes | 9 007 199 254 740 992 bits |
| 1 exabyte | 1 152 921 504 606 846 976 bytes | 9 223 372 036 854 775 808 bits |
| 1 zettabyte | 1 180 591 620 717 411 303 424 bytes | 9 444 732 965 739 290 427 392 bits |
| 1 yottabyte | 1 208 925 819 614 629 174 706 176 bytes | 9 671 406 556 917 033 397 649 408 bits |
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| Interesting Facts About Storage | ||
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