{"id":59,"date":"2023-06-05T15:29:39","date_gmt":"2023-06-05T15:29:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/putting-it-together-historical-counting-systems\/"},"modified":"2023-07-19T18:11:12","modified_gmt":"2023-07-19T18:11:12","slug":"putting-it-together-historical-counting-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/putting-it-together-historical-counting-systems\/","title":{"raw":"Putting It Together: Historical Counting Systems","rendered":"Putting It Together: Historical Counting Systems"},"content":{"raw":"\"The<\/a>Computers speak binary<\/strong>. \u00a0The binary, or base 2, positional number system uses only two digits, 0 and 1, which makes it ideal for computers whose basic components typically exist in two states, off<\/em> (0) or on<\/em> (1). \u00a0However, binary numbers can be very tedious for human readers to interpret. \u00a0For example, can you figure out what decimal number is represented by the following binary number?\r\n

[latex]110111101100000101011001000111111[\/latex]<\/p>\r\n \r\n\r\n \r\n\r\nThere are 33 digits in this numeral! \u00a0We may estimate roughly how big this number is by working out just the first few digits (we\u2019ll work out just four digits, but you could go further if you wanted to).\r\n

[latex]1 \\times 2^{32} + 1 \\times 2^{31} + 0 \\times 2^{30} + 1 \\times 2^{29}[\/latex]<\/p>\r\n

[latex]= 4,294,967,296 + 2,147,483,648 + 0 + 536,870,912[\/latex]<\/p>\r\n

[latex]\\approx 7,000,000,000[\/latex]<\/p>\r\nThis number, about 7 billion, represents the world population on January 1st, 2017 according to the website, Population.City.\r\n\r\n \r\n\r\nNow if we wanted to work out the decimal value of this number down to the last digit, there is a nice shortcut. \u00a0First let\u2019s separate the digits into groups of four.\r\n

[latex]1\\; 1011\\; 1101\\;1000\\; 0010\\; 1011\\; 0010\\; 0011\\; 1111[\/latex]<\/p>\r\nEach group may now be thought of as a numeral in base [latex]2^4 =16[\/latex]. \u00a0The base 16, or hexadecimal, number system requires 16 digits. \u00a0In particular, the digits corresponding to decimal 10 through 15 are usually written as the letters A through F, as indicated in Table I.\r\n\r\n \r\n\r\nTable I: \u00a0Hexadecimal digits and their equivalent decimal representations.\r\n

\r\n\r\n\r\n\r\n\r\n
Hexa-\r\n\r\ndecimal<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/td>\r\n4<\/td>\r\n5<\/td>\r\n6<\/td>\r\n7<\/td>\r\n8<\/td>\r\n9<\/td>\r\nA<\/td>\r\nB<\/td>\r\nC<\/td>\r\nD<\/td>\r\nE<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
Decimal<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/td>\r\n4<\/td>\r\n5<\/td>\r\n6<\/td>\r\n7<\/td>\r\n8<\/td>\r\n9<\/td>\r\n10<\/td>\r\n11<\/td>\r\n12<\/td>\r\n13<\/td>\r\n14<\/td>\r\n15<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nFurthermore, there is a one-to-one correspondence between every possible 4-digit binary expression and the hexadecimal digits, as shown in Table II. \u00a0It is this correspondence that makes hexadecimal a natural medium for writing large binary numbers in a more human-readable way.\r\n\r\n \r\n\r\nTable II: \u00a0Binary-hexadecimal conversion chart.\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Binary<\/td>\r\nHex.<\/td>\r\nBinary<\/td>\r\nHex.<\/td>\r\nBinary<\/td>\r\nHex.<\/td>\r\nBinary<\/td>\r\nHex.<\/td>\r\n<\/tr>\r\n
0000<\/td>\r\n0<\/td>\r\n0100<\/td>\r\n4<\/td>\r\n1000<\/td>\r\n8<\/td>\r\n1100<\/td>\r\nC<\/td>\r\n<\/tr>\r\n
0001<\/td>\r\n1<\/td>\r\n0101<\/td>\r\n5<\/td>\r\n1001<\/td>\r\n9<\/td>\r\n1101<\/td>\r\nD<\/td>\r\n<\/tr>\r\n
0010<\/td>\r\n2<\/td>\r\n0110<\/td>\r\n6<\/td>\r\n1010<\/td>\r\nA<\/td>\r\n1110<\/td>\r\nE<\/td>\r\n<\/tr>\r\n
0011<\/td>\r\n3<\/td>\r\n0111<\/td>\r\n7<\/td>\r\n1011<\/td>\r\nB<\/td>\r\n1111<\/td>\r\nF<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nSo let\u2019s see what the population of the Earth looks like in hexadecimal notation. \u00a0Note that the leading \u201c1\u201d in our numeral should be interpreted as \u201c0001.\u201d\r\n
\r\n\r\n\r\n\r\n\r\n
0001<\/td>\r\n1011<\/td>\r\n1101<\/td>\r\n1000<\/td>\r\n0010<\/td>\r\n1011<\/td>\r\n0010<\/td>\r\n0011<\/td>\r\n1111<\/td>\r\n<\/tr>\r\n
1<\/td>\r\nB<\/td>\r\nD<\/td>\r\n8<\/td>\r\n2<\/td>\r\nB<\/td>\r\n2<\/td>\r\n3<\/td>\r\nF<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nThe resulting hexadecimal number, 1BD82B23F, is much easier to write down and to work with than the original binary representation, at least for us non-computers.\r\n\r\n \r\n\r\nFinally, let\u2019s figure out the exact decimal representation of the number using what we have learned in this module about place-value systems.\r\n\r\n \r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Digit<\/td>\r\nPlace value<\/td>\r\nComputed values<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n[latex]16^8[\/latex]<\/td>\r\n[latex]1 \\times 16^8 = 4,294,967,296[\/latex]<\/td>\r\n<\/tr>\r\n
B<\/td>\r\n[latex]16^7[\/latex]<\/td>\r\n[latex]11 \\times 16^7 = 2,952,790,016[\/latex]<\/td>\r\n<\/tr>\r\n
D<\/td>\r\n[latex]16^6[\/latex]<\/td>\r\n[latex]13 \\times 16^6 = 218,103,808[\/latex]<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n[latex]16^5[\/latex]<\/td>\r\n[latex]8 \\times 16^5 = 8,388,608[\/latex]<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n[latex]16^4[\/latex]<\/td>\r\n[latex]2 \\times 16^4 = 131,072[\/latex]<\/td>\r\n<\/tr>\r\n
B<\/td>\r\n[latex]16^3[\/latex]<\/td>\r\n[latex]11 \\times 16^3 = 45,056[\/latex]<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n[latex]16^2[\/latex]<\/td>\r\n[latex]2 \\times 16^2 = 512[\/latex]<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n[latex]16[\/latex]<\/td>\r\n[latex]3 \\times 16 = 48[\/latex]<\/td>\r\n<\/tr>\r\n
F<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]15 \\times 1 = 15[\/latex]<\/td>\r\n<\/tr>\r\n
<\/td>\r\nTotal:<\/td>\r\n[latex]7,474,426,431[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nAlmost seven and a half billion people call this planet home. \u00a0That\u2019s a lot of people regardless whether the amount is expressed in decimal, hexadecimal, or binary notation.\r\n\r\nSo far in this unit, we have briefly sketched the development of numbers and our counting system, with the emphasis on the \u201cbrief\u201d part. There are numerous sources of information and research that fill many volumes of books on this topic. Unfortunately, we cannot begin to come close to covering all of the information that is out there.\r\n\r\n\"\"<\/a>\r\n\r\nWe have only scratched the surface of the wealth of research and information that exists on the development of numbers and counting throughout human history. What is important to note is that the system that we use every day is a product of thousands of years of progress and development. It represents contributions by many civilizations and cultures. It does not come down to us from the sky, a gift from the gods. It is not the creation of a textbook publisher. It is indeed as human as we are, as is the rest of mathematics. Behind every symbol, formula and rule there is a human face to be found, or at least sought.\r\n\r\nFurthermore, we hope that you now have a basic appreciation for just how interesting and diverse number systems can get. Also, we\u2019re pretty sure that you have also begun to recognize that we take our own number system for granted so much that when we try to adapt to other systems or bases, we find ourselves truly having to concentrate and think about what is going on.","rendered":"

\"The<\/a>Computers speak binary<\/strong>. \u00a0The binary, or base 2, positional number system uses only two digits, 0 and 1, which makes it ideal for computers whose basic components typically exist in two states, off<\/em> (0) or on<\/em> (1). \u00a0However, binary numbers can be very tedious for human readers to interpret. \u00a0For example, can you figure out what decimal number is represented by the following binary number?<\/p>\n

[latex]110111101100000101011001000111111[\/latex]<\/p>\n

 <\/p>\n

 <\/p>\n

There are 33 digits in this numeral! \u00a0We may estimate roughly how big this number is by working out just the first few digits (we\u2019ll work out just four digits, but you could go further if you wanted to).<\/p>\n

[latex]1 \\times 2^{32} + 1 \\times 2^{31} + 0 \\times 2^{30} + 1 \\times 2^{29}[\/latex]<\/p>\n

[latex]= 4,294,967,296 + 2,147,483,648 + 0 + 536,870,912[\/latex]<\/p>\n

[latex]\\approx 7,000,000,000[\/latex]<\/p>\n

This number, about 7 billion, represents the world population on January 1st, 2017 according to the website, Population.City.<\/p>\n

 <\/p>\n

Now if we wanted to work out the decimal value of this number down to the last digit, there is a nice shortcut. \u00a0First let\u2019s separate the digits into groups of four.<\/p>\n

[latex]1\\; 1011\\; 1101\\;1000\\; 0010\\; 1011\\; 0010\\; 0011\\; 1111[\/latex]<\/p>\n

Each group may now be thought of as a numeral in base [latex]2^4 =16[\/latex]. \u00a0The base 16, or hexadecimal, number system requires 16 digits. \u00a0In particular, the digits corresponding to decimal 10 through 15 are usually written as the letters A through F, as indicated in Table I.<\/p>\n

 <\/p>\n

Table I: \u00a0Hexadecimal digits and their equivalent decimal representations.<\/p>\n

\n\n\n\n\n
Hexa-<\/p>\n

decimal<\/td>\n

0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\nA<\/td>\nB<\/td>\nC<\/td>\nD<\/td>\nE<\/td>\nF<\/td>\n<\/tr>\n
Decimal<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n11<\/td>\n12<\/td>\n13<\/td>\n14<\/td>\n15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

 <\/p>\n

Furthermore, there is a one-to-one correspondence between every possible 4-digit binary expression and the hexadecimal digits, as shown in Table II. \u00a0It is this correspondence that makes hexadecimal a natural medium for writing large binary numbers in a more human-readable way.<\/p>\n

 <\/p>\n

Table II: \u00a0Binary-hexadecimal conversion chart.<\/p>\n

\n\n\n\n\n\n\n\n
Binary<\/td>\nHex.<\/td>\nBinary<\/td>\nHex.<\/td>\nBinary<\/td>\nHex.<\/td>\nBinary<\/td>\nHex.<\/td>\n<\/tr>\n
0000<\/td>\n0<\/td>\n0100<\/td>\n4<\/td>\n1000<\/td>\n8<\/td>\n1100<\/td>\nC<\/td>\n<\/tr>\n
0001<\/td>\n1<\/td>\n0101<\/td>\n5<\/td>\n1001<\/td>\n9<\/td>\n1101<\/td>\nD<\/td>\n<\/tr>\n
0010<\/td>\n2<\/td>\n0110<\/td>\n6<\/td>\n1010<\/td>\nA<\/td>\n1110<\/td>\nE<\/td>\n<\/tr>\n
0011<\/td>\n3<\/td>\n0111<\/td>\n7<\/td>\n1011<\/td>\nB<\/td>\n1111<\/td>\nF<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

 <\/p>\n

So let\u2019s see what the population of the Earth looks like in hexadecimal notation. \u00a0Note that the leading \u201c1\u201d in our numeral should be interpreted as \u201c0001.\u201d<\/p>\n

\n\n\n\n\n
0001<\/td>\n1011<\/td>\n1101<\/td>\n1000<\/td>\n0010<\/td>\n1011<\/td>\n0010<\/td>\n0011<\/td>\n1111<\/td>\n<\/tr>\n
1<\/td>\nB<\/td>\nD<\/td>\n8<\/td>\n2<\/td>\nB<\/td>\n2<\/td>\n3<\/td>\nF<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

 <\/p>\n

The resulting hexadecimal number, 1BD82B23F, is much easier to write down and to work with than the original binary representation, at least for us non-computers.<\/p>\n

 <\/p>\n

Finally, let\u2019s figure out the exact decimal representation of the number using what we have learned in this module about place-value systems.<\/p>\n

 <\/p>\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Digit<\/td>\nPlace value<\/td>\nComputed values<\/td>\n<\/tr>\n
1<\/td>\n[latex]16^8[\/latex]<\/td>\n[latex]1 \\times 16^8 = 4,294,967,296[\/latex]<\/td>\n<\/tr>\n
B<\/td>\n[latex]16^7[\/latex]<\/td>\n[latex]11 \\times 16^7 = 2,952,790,016[\/latex]<\/td>\n<\/tr>\n
D<\/td>\n[latex]16^6[\/latex]<\/td>\n[latex]13 \\times 16^6 = 218,103,808[\/latex]<\/td>\n<\/tr>\n
8<\/td>\n[latex]16^5[\/latex]<\/td>\n[latex]8 \\times 16^5 = 8,388,608[\/latex]<\/td>\n<\/tr>\n
2<\/td>\n[latex]16^4[\/latex]<\/td>\n[latex]2 \\times 16^4 = 131,072[\/latex]<\/td>\n<\/tr>\n
B<\/td>\n[latex]16^3[\/latex]<\/td>\n[latex]11 \\times 16^3 = 45,056[\/latex]<\/td>\n<\/tr>\n
2<\/td>\n[latex]16^2[\/latex]<\/td>\n[latex]2 \\times 16^2 = 512[\/latex]<\/td>\n<\/tr>\n
3<\/td>\n[latex]16[\/latex]<\/td>\n[latex]3 \\times 16 = 48[\/latex]<\/td>\n<\/tr>\n
F<\/td>\n[latex]1[\/latex]<\/td>\n[latex]15 \\times 1 = 15[\/latex]<\/td>\n<\/tr>\n
<\/td>\nTotal:<\/td>\n[latex]7,474,426,431[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

 <\/p>\n

Almost seven and a half billion people call this planet home. \u00a0That\u2019s a lot of people regardless whether the amount is expressed in decimal, hexadecimal, or binary notation.<\/p>\n

So far in this unit, we have briefly sketched the development of numbers and our counting system, with the emphasis on the \u201cbrief\u201d part. There are numerous sources of information and research that fill many volumes of books on this topic. Unfortunately, we cannot begin to come close to covering all of the information that is out there.<\/p>\n

\"\"<\/a><\/p>\n

We have only scratched the surface of the wealth of research and information that exists on the development of numbers and counting throughout human history. What is important to note is that the system that we use every day is a product of thousands of years of progress and development. It represents contributions by many civilizations and cultures. It does not come down to us from the sky, a gift from the gods. It is not the creation of a textbook publisher. It is indeed as human as we are, as is the rest of mathematics. Behind every symbol, formula and rule there is a human face to be found, or at least sought.<\/p>\n

Furthermore, we hope that you now have a basic appreciation for just how interesting and diverse number systems can get. Also, we\u2019re pretty sure that you have also begun to recognize that we take our own number system for granted so much that when we try to adapt to other systems or bases, we find ourselves truly having to concentrate and think about what is going on.<\/p>\n\n\t\t\t

\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Original<\/div>