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Base Ten Number System Our Base Ten Number System Is Called the Hindu-Arabic System

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Base Ten Number System Our Base Ten Number System Is Called the Hindu-Arabic System Base Ten Number System Our base ten number system is called the Hindu-Arabic system. In this system there are ten different symbols (digits) to represent quantities, 0,1,2,3,4,5,6,7,8,9. In fact, there are number systems in use that do this, namely the binary (two symbols 0 and 1), octal (eight symbols 0,1,2,3,4,5,6,7) and hexadecimal (sixteen different symbols, 0-9,A,B,C,D,E,F). How Did Numbers Arise? To understand how number systems work, let us first imagine that there we are cavepeople and that there is no number system. The only way we could represent how many objects there are in a set would be to write something like Of course this system will become very cumbersome as the number of objects begins increasing. The first simplification is to start grouping objects or placing a symbol when we get to a certain quantity. The number of objects which we group is completely arbitrary. 1 •We need symbols to represent objects up to the point that we group them. •Let us imagine that we group whenever we have six objects, i.e. we have six different symbols •They are 0,1,2,3,4,5 (none) =0 • I = 1 • II = 2 • III = 3 • IIII =4 • IIIII =5 If we add one more object we must group the objects we have and begin our numbering from the beginning specifying the first digit as a number of groups of six and the second as how many ungouped objects left over. • IIIIII = 1 0 • IIIIII I = 1 1 • IIIIII I I = 1 2 • IIIIII III =1 3 • IIIIII IIII =1 4 • IIIIII IIIII =1 5 • IIIIII IIIIII =2 0 • IIIIII IIIIII I =2 1 • IIIIII IIIIII II=2 2 2 Eventually we will have = 5 5 If we add one more object We write these as 1 0 0 =100 The system is called a place-value number system because each place or position of a digit determines it’s value. The Number of different digits is called the base. For example in base six the values are: Groups Groups Ones Of six- of sixes sixes (Thirty six) A number written in base six would look like Total=105 2 5 3 Representing 2 thirty sixes, five sixes, and three ones 3 Going the other way-imagine you had 81 objects, how would you write this number in base six? Groups Groups Ones Of six- of sixes 81 =213 sixes 10 6 (Thirty six) Out of the 81, how many groups of thirty sixes can you make? How may groups of sixes ? How many ones left over? •You can make Two groups of thirty sixes (72 leaving 9) •Out of the nine you can make one group of six (Leaving 3) •You have three left over. So 81 in base ten is written in base six Our Base Ten Number System Tens Ones H u n d r e d s Tens Ones 4 The Egyptians had a number system using seven different symbols. 1 is shown by a single stroke. 10 is shown by a drawing of a hobble for cattle. 100 is represented by a coil of rope. 1,000 is a drawing of a lotus plant. 10,000 is represented by a finger. 100,000 by a tadpole or frog 1,000,000 is the figure of a god with arms raised above his head. 5 Mayan Number System The Mayans devised a counting system that was able to represent very large numbers by using only 3 symbols, a dot, a bar, and a symbol for zero, or completion, usually a shell. 6 Arabic Numerals Here is an example of an early form of Indian numerals being used in the eastern part of the Arabic empire around first century AD 969 A.D. Binary Numbers Computers happen to operate using the base-2 number system, also known as the binary number system. The word bit is a shortening of the words "Binary digit." Whereas decimal digits have 10 possible values ranging from 0 to 9, bits have only two possible values: 0 and 1. Therefore, a binary number is composed of only 0s and 1s, like this: 1011. How do you figure out what the value of the binary number 1011 is? You can see that in binary numbers, each bit holds the value of increasing powers of 2. The reason each position’s value is a multiple of two is because there are only two symbols. 7 We can look at the binary system as a number of Off/On switches. If the switch is ‘On’ this represents a 1, if it is ‘Off’ it represents a zero. If all the swtches are ‘On’ in a 8 digit number then the number represented is 255 (Add 128+64+32+16+8+4+2+1) Each digit is called a bit. On the right you see the representation for the number 58. (32+16+8+2) This is how you write numbers in binary 0 = 0 10 = 1010 1 = 1 11 = 1011 2 = 10 12 = 1100 3 = 11 13 = 1101 4 = 100 14 = 1110 5 = 101 15 = 1111 6 = 110 16 = 10000 7 = 111 17 = 10001 8 = 1000 18 = 10010 9 = 1001 19 = 10011 20 = 10100 8.
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