Binary and Octal
In 1679, Gottfried Leibniz invented the binary numeral system, and published the first information about it in his article "Explication de l'Arithmétique Binaire". Gottfried, born 7/1/1646, was a prominent German philosopher and mathematician. He conducted work that is still studied today, work on calculus, mathematical theory, mechanical calculators, and the binary system.
Binary coding is based off of this system, and it is used in almost all modern computers. The binary numeral system is a system that uses 2 bases, or 2 symbols that express numerical values. These symbols are 0 and 1. Every digit representing one value is called a bit. Since the only digits are 1 and 0, the highest value that is represented singularly by its own digit is 1. 0 in the base 10 system equals 0, 1 equals 1, then 2 is represented by 10. To go up one value, all you do is add 1 digit to the next place over (in the base ten system, the places would be tens, hundreds, thousands, e.t.c.) and replace the former column, or place,with a 0, except when a zero is already located there. In that case, you add a 1. For example: 0=0, 1=1, 2=10, 3=11, 4=100, 5=101, 6=110, and 7=111. Counting in order works well, but only with small numbers. With larger numbers, it would take too long to count it out in your head from 0. What helps when converting from base 2 to base 10, or vice versa, is to remember that every digit in a binary number represents a power of 2. The numbering of the binary number 111111 is 1-6, since the binary number has six digits. To convert the binary number into a base 10 number, you take the digit and multiply it times 2 to the power of whichever number digit it is. The 1 farthest to the left is the 6th digit, so you would multiply 1 times 2 to the power of 6. You do so with every digit until you finish, and then you add all of the products together. The binary number will convert into a base 10 number.
Converting from base 10 to base 2 is a whole other process. Earlier, I already explained how to convert small numbers from base 10 to base 2. To convert larger numbers, you have to use division. You take the number in base 10 you want to convert, for example, 357, and divide it evenly by 2, keeping track of the remainders of each division:
Once you are finished, you begin at the top and read the remainders as a binary number: 101100101. From what was demonstrated above, you can see the base 10 number 357 equals the base 2 number 101100101, or 35710 equals 1011001012.
The octal numeral system works the same way as the base 10 and the binary system, except it has a base made of eight numbers. The eight digits of the octal system are 0, 1, 2, 3, 4, 5, and 7. The octal system was first discovered in 1716 by Emanuel Swedenborg, although linguists speculate that it was discovered by the Proto-Indo-Europeans, a group of Neolithic people ancestors to the Indo-European people of the Bronze Age. However, in 1716 King Charles XII of Sweden commanded Emanuel Swedenborg to develop a new numeral system based on 64 that would replace the decimal numeral system. Emanuel Swedenborg argued against 64 and instead proposed a base 8 system, saying that the base 64 system would be too difficult for people less intelligent than the king to comprehend. Swedenborg wrote a paper on this new system in Swedish, where he wrote the numbers 1 through 7 as the consonants l, s, n, m, t, f, and u (v), with zero written as the vowel o. With this mechanism, 8=lo, 16=so, 24=no, and 64=loo. At the time there was a special rule in place that interjected vowels in between consecutive consonants. After Emanuel Swedenborg’s “discovery” of the octal numeral system, many other mathematicians continued to ponder the octal numeral system.
As it has more base numbers than the binary system, the octal system is helpful for compacting binary numbers and making them more manageable. In the beginning of computers, it was a popular numeral system because it did this, and also because inputs and outputs for a computer were grouped together by eights, or by bytes. As computers were modernized, the octal system became less and less popular and the hexadecimal more so.
Counting in octal goes like this: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, and so on. You count normally up to 7, and then you add 1 to the next column, or place. While 10 in octal form appears to be ten in base 10 form, it is actually 8 in base 10 form. This means that after 7, 10 in octal equals 8 in decimal, 11=9, 12=10, 13=11, and so on.
To convert from octal to decimal, you use a similar method as when you convert from binary to decimal. In binary, each place represents a power of 2, while in octal, each place represents a power of 8. If my number is 3878 (in octal) , then it equals 7 times 8 to the power of 0 plus 8 times 8 to the power of 1 plus 3 times 8 to the power of 2. Once you work it out, this ends up being 263 in base 10. Here is the calculation: (7x80) + (8x81) + (3x82)= (7)+(64)+(3x64)=71+192=26310.
To convert from decimal to octal, you take the number in base 10, for example 891, and continuously divide it evenly by 8, keeping track of all of the remainders. 891 divided by 8 equals 111, with a remainder of 3. 111 divided by 8 equals 13, with a remainder of 7. 13 divided by 8 equals 1, with a remainder of 5. One is a number less than eight so you keep track of that as well. Placing the last quotient farthest to the left, you arrange the remainders into our octal number: 1573.
Finally, to convert from binary to octal and back again, you take the binary number 010101 and arrange it into groups of 3: 010 and 101. Thinking back to how you count small numbers in binary, you interpret each group as one decimal number. The binary number 010 equals 2 in decimal, and the binary number 101 equals 5 in decimal. The find the octal number, you read the decimal numbers as one number: 25. In order to convert from octal back to binary, you think through the same process backwards. You take your octal number, 25, and interpret each digit as a decimal number that equals a binary number: 2 equals 010 and 5 equals 101. To find the binary number, you just put these two small binary numbers next to each other and you have 10101.
Sources:
http://homepages.math.uic.edu/~jlewis/mtht430/chap7b.pdf
http://www.purplemath.com/modules/numbbase.htm
http://www.electronics-tutorials.ws/binary/bin_4.html
http://en.wikipedia.org/wiki/Hexadecimal
http://en.wikipedia.org/wiki/Octal
http://www.electrical4u.com/binary-to-octal-and-octal-to-binary-conversion/
Binary coding is based off of this system, and it is used in almost all modern computers. The binary numeral system is a system that uses 2 bases, or 2 symbols that express numerical values. These symbols are 0 and 1. Every digit representing one value is called a bit. Since the only digits are 1 and 0, the highest value that is represented singularly by its own digit is 1. 0 in the base 10 system equals 0, 1 equals 1, then 2 is represented by 10. To go up one value, all you do is add 1 digit to the next place over (in the base ten system, the places would be tens, hundreds, thousands, e.t.c.) and replace the former column, or place,with a 0, except when a zero is already located there. In that case, you add a 1. For example: 0=0, 1=1, 2=10, 3=11, 4=100, 5=101, 6=110, and 7=111. Counting in order works well, but only with small numbers. With larger numbers, it would take too long to count it out in your head from 0. What helps when converting from base 2 to base 10, or vice versa, is to remember that every digit in a binary number represents a power of 2. The numbering of the binary number 111111 is 1-6, since the binary number has six digits. To convert the binary number into a base 10 number, you take the digit and multiply it times 2 to the power of whichever number digit it is. The 1 farthest to the left is the 6th digit, so you would multiply 1 times 2 to the power of 6. You do so with every digit until you finish, and then you add all of the products together. The binary number will convert into a base 10 number.
Converting from base 10 to base 2 is a whole other process. Earlier, I already explained how to convert small numbers from base 10 to base 2. To convert larger numbers, you have to use division. You take the number in base 10 you want to convert, for example, 357, and divide it evenly by 2, keeping track of the remainders of each division:
Once you are finished, you begin at the top and read the remainders as a binary number: 101100101. From what was demonstrated above, you can see the base 10 number 357 equals the base 2 number 101100101, or 35710 equals 1011001012.
The octal numeral system works the same way as the base 10 and the binary system, except it has a base made of eight numbers. The eight digits of the octal system are 0, 1, 2, 3, 4, 5, and 7. The octal system was first discovered in 1716 by Emanuel Swedenborg, although linguists speculate that it was discovered by the Proto-Indo-Europeans, a group of Neolithic people ancestors to the Indo-European people of the Bronze Age. However, in 1716 King Charles XII of Sweden commanded Emanuel Swedenborg to develop a new numeral system based on 64 that would replace the decimal numeral system. Emanuel Swedenborg argued against 64 and instead proposed a base 8 system, saying that the base 64 system would be too difficult for people less intelligent than the king to comprehend. Swedenborg wrote a paper on this new system in Swedish, where he wrote the numbers 1 through 7 as the consonants l, s, n, m, t, f, and u (v), with zero written as the vowel o. With this mechanism, 8=lo, 16=so, 24=no, and 64=loo. At the time there was a special rule in place that interjected vowels in between consecutive consonants. After Emanuel Swedenborg’s “discovery” of the octal numeral system, many other mathematicians continued to ponder the octal numeral system.
As it has more base numbers than the binary system, the octal system is helpful for compacting binary numbers and making them more manageable. In the beginning of computers, it was a popular numeral system because it did this, and also because inputs and outputs for a computer were grouped together by eights, or by bytes. As computers were modernized, the octal system became less and less popular and the hexadecimal more so.
Counting in octal goes like this: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, and so on. You count normally up to 7, and then you add 1 to the next column, or place. While 10 in octal form appears to be ten in base 10 form, it is actually 8 in base 10 form. This means that after 7, 10 in octal equals 8 in decimal, 11=9, 12=10, 13=11, and so on.
To convert from octal to decimal, you use a similar method as when you convert from binary to decimal. In binary, each place represents a power of 2, while in octal, each place represents a power of 8. If my number is 3878 (in octal) , then it equals 7 times 8 to the power of 0 plus 8 times 8 to the power of 1 plus 3 times 8 to the power of 2. Once you work it out, this ends up being 263 in base 10. Here is the calculation: (7x80) + (8x81) + (3x82)= (7)+(64)+(3x64)=71+192=26310.
To convert from decimal to octal, you take the number in base 10, for example 891, and continuously divide it evenly by 8, keeping track of all of the remainders. 891 divided by 8 equals 111, with a remainder of 3. 111 divided by 8 equals 13, with a remainder of 7. 13 divided by 8 equals 1, with a remainder of 5. One is a number less than eight so you keep track of that as well. Placing the last quotient farthest to the left, you arrange the remainders into our octal number: 1573.
Finally, to convert from binary to octal and back again, you take the binary number 010101 and arrange it into groups of 3: 010 and 101. Thinking back to how you count small numbers in binary, you interpret each group as one decimal number. The binary number 010 equals 2 in decimal, and the binary number 101 equals 5 in decimal. The find the octal number, you read the decimal numbers as one number: 25. In order to convert from octal back to binary, you think through the same process backwards. You take your octal number, 25, and interpret each digit as a decimal number that equals a binary number: 2 equals 010 and 5 equals 101. To find the binary number, you just put these two small binary numbers next to each other and you have 10101.
Sources:
http://homepages.math.uic.edu/~jlewis/mtht430/chap7b.pdf
http://www.purplemath.com/modules/numbbase.htm
http://www.electronics-tutorials.ws/binary/bin_4.html
http://en.wikipedia.org/wiki/Hexadecimal
http://en.wikipedia.org/wiki/Octal
http://www.electrical4u.com/binary-to-octal-and-octal-to-binary-conversion/