This is a very interesting topic and is useful to know.
Unary is base 1 and it isn't very efficient but it is used when you count or tally up things. Not quite as useful as the other bases. adding and subtracting is easy in unary but if everything is written as a string of 1s in unary how would you multiply or divide? what about 0 or negatives?
Binary is twice as efficient as unary and is used a lot in computer programming. Even my typing is in binary but you don't see 10001001011110101 or something like that, you see letters and other symbols. adding and subtracting is easy here as well except when you carry and you carry on to the last digit you can't just write 230 or something like that, same with borrowing.
Ternary is used in some very expensive computers and just like how 1 byte is 8 bits or binary digits there is a tryte which is 6 trits and a trit is a ternary digit.
Quarternary isn't used often just like base 3 isn't but when it is used it is used for 2D Hilbert curves and another way of writing nucleotide sequences in genetics
Quinary or base 5 isn't used very often either and it often a subbase of base 10 such as in a Japanese abacus.
Senary or base 6 is useful for studying primes since all primes except for 2 and 3 have in the ones place either a 1 or 5 in base 6.
Base 7 isn't used often.
Octal or base 8 is used often in computers just like binary is.
Base 9 has a lot of matematical characteristics.
Except for three, no primes in nonary end in 0, 3 or 6, since any nonary number ending in 0, 3 or 6 is divisible by three.
A nonary number is divisible by two, four, or eight if the sum of its digits is also divisible by two, four, or eight respectively.
If x is a triangular number, so is 9x+1.This means that one finds 3, 31, 311, 3111, 31111... in the triangular numbers. Likewise, 6, 61, 611, 6111, ....
Nonary is useful for determining the sum of the sum of all numbers in a sequence's digits until a single digit is obtained. For example if one was to determine the sum of all digits in the number 382, the result would be found by 3+8+2=13 however this number has more than one digit, so the process continues, 1+3=4 therefore the number 382 would solve to be 4. This answer may be found more easily with Nonary by simply converting 382 into the base 9, which gives 464, the last digit of which will always be the result found by adding each digit up until a single digit is achieved, where 0 reflects the answer of 9.
Decimal or base 10 is the most common base used. Just about everything we do in math is base 10 except for that digital root thing in which you use base 9 instead or programming where you use base 2.
base 11 isn't used often. The check digit for ISBN is found as the result of taking modulo 11. Since this could give 11 possible results, the digit "X" is used in place of "10".
base 12 just like base 6 has terminating decimals only if the denominator has these factors: 2 and/or 3.
base 13 is used in the mayan calendar. In the end of The Restaurant at the End of the Universe by Douglas Adams, a possible question to get the answer "forty-two" is presented: "What do you get if you multiply six by nine?" Of course, the answer is deliberately wrong, creating a humorous effect – if the calculation is carried out in base 10. People who were trying to find a deeper meaning in the passage soon noticed that in base 13, 6 sub 13 × 9 sub 13 is actually 42 sub 13 (as 4 × 13 + 2 = 54, i.e. 54 in decimal is equal to 42 expressed in base 13). When confronted with this, the author claimed that it was a mere coincidence, famously stating that "I may be a sorry case, but I don't write jokes in base 13." (However, as observed in the Annotated Alice, one of Lewis Carroll's most elaborate jokes featuring the number 42 is a calculation in chapter 2 of Alice's Adventures in Wonderland that breaks down precisely when one multiplies by 13 and attempts to express the answer in base 42.)
base 14 isn't common.
Neither is base 15.
Hexadecimal is very common and is used in RGB to determine the exact color.
Mayan numerals are in base 20.
solar time includes a base 24 component
Base 26 isn't common.
Septemvigesimal(base 27) notation can be used as a concise representation of ternary data, where each septemvigesimal digit represents three ternary digits. This is similar to using octal notation to represent binary data,[3] though the digit set is closer in size to hexadecimal.
base 32 isn't common.
base 36 can be interpreted in lots of programming languages.
base 60 is used in time and angles.
now in all of these we can represent numbers and do operations but for example how would you multiply 100 in base 60(60^2 + 0 * 60^1 + 0 * 60^0) by 100 in base 60? and how would you convert that to base 10?
Unary is base 1 and it isn't very efficient but it is used when you count or tally up things. Not quite as useful as the other bases. adding and subtracting is easy in unary but if everything is written as a string of 1s in unary how would you multiply or divide? what about 0 or negatives?
Binary is twice as efficient as unary and is used a lot in computer programming. Even my typing is in binary but you don't see 10001001011110101 or something like that, you see letters and other symbols. adding and subtracting is easy here as well except when you carry and you carry on to the last digit you can't just write 230 or something like that, same with borrowing.
Ternary is used in some very expensive computers and just like how 1 byte is 8 bits or binary digits there is a tryte which is 6 trits and a trit is a ternary digit.
Quarternary isn't used often just like base 3 isn't but when it is used it is used for 2D Hilbert curves and another way of writing nucleotide sequences in genetics
Quinary or base 5 isn't used very often either and it often a subbase of base 10 such as in a Japanese abacus.
Senary or base 6 is useful for studying primes since all primes except for 2 and 3 have in the ones place either a 1 or 5 in base 6.
Base 7 isn't used often.
Octal or base 8 is used often in computers just like binary is.
Base 9 has a lot of matematical characteristics.
Except for three, no primes in nonary end in 0, 3 or 6, since any nonary number ending in 0, 3 or 6 is divisible by three.
A nonary number is divisible by two, four, or eight if the sum of its digits is also divisible by two, four, or eight respectively.
If x is a triangular number, so is 9x+1.This means that one finds 3, 31, 311, 3111, 31111... in the triangular numbers. Likewise, 6, 61, 611, 6111, ....
Nonary is useful for determining the sum of the sum of all numbers in a sequence's digits until a single digit is obtained. For example if one was to determine the sum of all digits in the number 382, the result would be found by 3+8+2=13 however this number has more than one digit, so the process continues, 1+3=4 therefore the number 382 would solve to be 4. This answer may be found more easily with Nonary by simply converting 382 into the base 9, which gives 464, the last digit of which will always be the result found by adding each digit up until a single digit is achieved, where 0 reflects the answer of 9.
Decimal or base 10 is the most common base used. Just about everything we do in math is base 10 except for that digital root thing in which you use base 9 instead or programming where you use base 2.
base 11 isn't used often. The check digit for ISBN is found as the result of taking modulo 11. Since this could give 11 possible results, the digit "X" is used in place of "10".
base 12 just like base 6 has terminating decimals only if the denominator has these factors: 2 and/or 3.
base 13 is used in the mayan calendar. In the end of The Restaurant at the End of the Universe by Douglas Adams, a possible question to get the answer "forty-two" is presented: "What do you get if you multiply six by nine?" Of course, the answer is deliberately wrong, creating a humorous effect – if the calculation is carried out in base 10. People who were trying to find a deeper meaning in the passage soon noticed that in base 13, 6 sub 13 × 9 sub 13 is actually 42 sub 13 (as 4 × 13 + 2 = 54, i.e. 54 in decimal is equal to 42 expressed in base 13). When confronted with this, the author claimed that it was a mere coincidence, famously stating that "I may be a sorry case, but I don't write jokes in base 13." (However, as observed in the Annotated Alice, one of Lewis Carroll's most elaborate jokes featuring the number 42 is a calculation in chapter 2 of Alice's Adventures in Wonderland that breaks down precisely when one multiplies by 13 and attempts to express the answer in base 42.)
base 14 isn't common.
Neither is base 15.
Hexadecimal is very common and is used in RGB to determine the exact color.
Mayan numerals are in base 20.
solar time includes a base 24 component
Base 26 isn't common.
Septemvigesimal(base 27) notation can be used as a concise representation of ternary data, where each septemvigesimal digit represents three ternary digits. This is similar to using octal notation to represent binary data,[3] though the digit set is closer in size to hexadecimal.
base 32 isn't common.
base 36 can be interpreted in lots of programming languages.
base 60 is used in time and angles.
now in all of these we can represent numbers and do operations but for example how would you multiply 100 in base 60(60^2 + 0 * 60^1 + 0 * 60^0) by 100 in base 60? and how would you convert that to base 10?