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HOW TO SEE NUMERICAL SYSTEMS We can build infinite numerical systems. What change is the basis. The most known are decimal , binary and hexadecimal In decimal system(the usual one )the basis is the number 10 (we have ten fingers) It is assumed in math that any number powered to zero equals 1 So you have in your mind the term “powered to” 2^4=2x2x2x2=16 5^3=5x5X5=125 …. Now we go how to build numerical systems starting with decimal system: In decimal system we have ten symbols 1,2 ,3 ,4 , 5 , 6 , 7 , 8 , 9 ,0 In binrary we have only 2 symbols 1, 0 In hexadecimal 16 simbols 1, 2 , 3 , 4 , 5 , 6 , 7 , , 8 , 9 , 0 , A, B, C, D, E, F Returning to decimal system Number examples expressed in its basis (in this case 10) 16=1x10^1x+6x10^0 (1 is the first digit, 10 is the basis, 6 is the second digit 32=3x10^1+2x10^0 120=1x10^2+2x\10^1+0^0 3024= 3x10^3+0x10^2+2x10^1+4x10^0 12560= 1x10^4+2x10^3+5x10^2+6x10^1+0^0 Numbers in bold are the power counted from left to right of the digit position beging in zero And what about binary system ? We only have two digits 1 , 0 How can we write any number in a binary system basis? It is the same logical structure! Binary System Unlike the decimal system, only two digits - 0, 1 - suffice to represent a number in the binary system. The binary system plays a crucial role in computer science and technology. The first 20 numbers in the binary notation are 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, the origin of which may be better understood if they are re-written in the following way: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 11: 01011 12: 01100 13: 01101 14: 01110 15: 01111 16: 10000 17: 10001 18: 10010 19: 10011 20: 10100 Lets see some binary numbers and convert them to decimal 1000 = 1x2^3+0x2^2+0x2^1+0x0^0=8+0+0+0=8 1111=1x2^3+1x2^2+1x2^1+1x2^0=8+4+2+1=8+4+1=15 Or using the digits written like above Any binary number can be converted into thedecimal system by summing the appropriate multiples of the different powers of two. For example, starting from the right, 10101101 represents (1 x 20) + (0 x 21) + (1 x 22) + (1 x 23) + (0 x 24) + (1 x 25) + (0 x 26) + (1 x 27) = 173. This example can be used for the conversion of binary numbers into decimal numbers. For the conversion of decimal numbers to binary numbers, the same principle can be used, but the other way around. Thus, to convert, first find the highest power of two that does not exceed the given number, and place a 1 in the corresponding position in the binary number. For example, the highest power of two in the decimal number 519 is 29 = 512. Thus, a 1 can be inserted as the 10th digit, counted from the right: 1000000000. In the remainder, 519 - 512 = 7, the highest power of 2 is 22 = 4, so the third zero from the right can be replaced by a 1: 1000000100. The next remainder, 3, consists of the sum of two powers of 2: 21 + 20, so the first and second zeros from the right are replaced by 1: 519 = 10000001112.