Post by microfarad on May 30, 2010 16:32:29 GMT
We use 10 little digits to make numbers. The digits we use are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
We put numbers in different columns to denote a multiple of their value. We use the multiples 1, 10, 100, 1000, 10000, etc. for our numbers. These values are powers of 10 (10^x). For example, the number 1295 has a 2 in the 100 column, and a 5 in the 1 column. The whole number, in expanded notation is:
1*1000 + 2*100 + 9*10 + 5*1
Our system is called base 10 because we use powers of ten for our columns and we use 10 digits. Base 3 uses 3 digits, and powers of 3. So to count in base three we go:
0, 1, 2
Wait... What do do now? We can't go to 3, it only uses the digits 0-2!!! Well, just like when we reach 9 in decimal, we have to increment the next column...
10, 11, 12
And increment the next column again...
20, 21, 22
But... What happens now? Ah, it increments the column AFTER
100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222
And...
1000, 1001, 1002, 1010, 1011, 1012, 1020...
But what do the numbers mean? Well, each column is a power of 3. So the columns are:
1, 3, 9, 27...
If we had the base 3 number 121, we could write it in expanded notation as...
1*9 + 2*3 + 1*1
which equals...
16 in base 10!
But what about that number system called binary? Binary is base 2, uses the digits 0 and 1, and uses columns with powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048...)
Thus counting in binary...
1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011...
Binary is commonly stereotyped because computers can easily store the digits 0 and 1, thus making extensive use of binary. Infortunately, this leads some people to believe that binary MUST have leading zeros, as is the practice when it is used for programming computers...
00000001, 00000010, 00000011, 00000100, 00000101, 00000110, 00000111, 00001000...
While this is more useful for storing in computers, binary is not mutually connected to those leading zeros. It's kike putting leading zeros on our system...
00000001, 00000002, 00000003, 00000004, 00000005, 00000006, 00000007, 00000008, 00000009, 00000010, 00000011, 00000012...
If you need more digits than just 0-9, you need to start using capital letters. Hexadecimal (base 16) is a common number system that uses the digits 0-F
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20...
When writing in expanded notation for conversion to our system, be sure to assign the appropriate number values to the letters. For example...
A=10, B=11, C=12, D=13...
Since hex numbers (hexadecimal numbers) us the columns 1, 16, 256... The number B4F is...
11*256 + 4*16 + 15*1
which is...
2847 in base 10
Other systems have decimals too. We use the column values of .1, .01, .001 after the decimal place. These are just 10 to a negative power. So, column values for other systems extend into the negative as well. Take base 5 as an example
5^3, 5^2, 5^1, 5^0, 5^-1, 5^-2, 5^-3
which is in our system (base 10)
125, 25, 5, 1, .2, .04, .008
Have fun with your new knowledge!
We put numbers in different columns to denote a multiple of their value. We use the multiples 1, 10, 100, 1000, 10000, etc. for our numbers. These values are powers of 10 (10^x). For example, the number 1295 has a 2 in the 100 column, and a 5 in the 1 column. The whole number, in expanded notation is:
1*1000 + 2*100 + 9*10 + 5*1
Our system is called base 10 because we use powers of ten for our columns and we use 10 digits. Base 3 uses 3 digits, and powers of 3. So to count in base three we go:
0, 1, 2
Wait... What do do now? We can't go to 3, it only uses the digits 0-2!!! Well, just like when we reach 9 in decimal, we have to increment the next column...
10, 11, 12
And increment the next column again...
20, 21, 22
But... What happens now? Ah, it increments the column AFTER
100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222
And...
1000, 1001, 1002, 1010, 1011, 1012, 1020...
But what do the numbers mean? Well, each column is a power of 3. So the columns are:
1, 3, 9, 27...
If we had the base 3 number 121, we could write it in expanded notation as...
1*9 + 2*3 + 1*1
which equals...
16 in base 10!
But what about that number system called binary? Binary is base 2, uses the digits 0 and 1, and uses columns with powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048...)
Thus counting in binary...
1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011...
Binary is commonly stereotyped because computers can easily store the digits 0 and 1, thus making extensive use of binary. Infortunately, this leads some people to believe that binary MUST have leading zeros, as is the practice when it is used for programming computers...
00000001, 00000010, 00000011, 00000100, 00000101, 00000110, 00000111, 00001000...
While this is more useful for storing in computers, binary is not mutually connected to those leading zeros. It's kike putting leading zeros on our system...
00000001, 00000002, 00000003, 00000004, 00000005, 00000006, 00000007, 00000008, 00000009, 00000010, 00000011, 00000012...
If you need more digits than just 0-9, you need to start using capital letters. Hexadecimal (base 16) is a common number system that uses the digits 0-F
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20...
When writing in expanded notation for conversion to our system, be sure to assign the appropriate number values to the letters. For example...
A=10, B=11, C=12, D=13...
Since hex numbers (hexadecimal numbers) us the columns 1, 16, 256... The number B4F is...
11*256 + 4*16 + 15*1
which is...
2847 in base 10
Other systems have decimals too. We use the column values of .1, .01, .001 after the decimal place. These are just 10 to a negative power. So, column values for other systems extend into the negative as well. Take base 5 as an example
5^3, 5^2, 5^1, 5^0, 5^-1, 5^-2, 5^-3
which is in our system (base 10)
125, 25, 5, 1, .2, .04, .008
Have fun with your new knowledge!