Some questions about bases and the decimal system.

[Logical Dog]

Hello.

This is how every number in the decimal system is expressed:

I had understood this topic earlier but as I was revising it today I have become confused somewhat.

I know that for the decimal system, we have 9 digits.

I understand this:

- When we use a base between 1-10, we do not need to come up with new digits.

- in the illustration one above, if the base is x, the coefficient a can only be less than x? Am I correct?

- I am confused as to how to divide numbers and change them to other bases, I seem to have forgotten basic division.

I understand 372 expressed in base 10 is 3*10*10 +7*10 + 2

But what I don't get is this, 372 is already expressed in base 10 when we divide it by 10?

I am really confused. All numbers are intuitively expressed in base 10 before we convert them to other bases..but division by the same base in the same form seems to give the same number.

 

[Logical Dog]

in the book I am using there is an example using 372, the author divides by 10 to reinforce the point about how the coefficients a an-1 an-2 are also the remainders of succcesful division.

But, 372 is already expressed in base 10 form when he divides, so, what is the point>

 

[mathman]

in the book I am using there is an example using 372, the author divides by 10 to reinforce the point about how the coefficients a an-1 an-2 are also the remainders of succcesful division.

But, 372 is already expressed in base 10 form when he divides, so, what is the point>

The point is to illustrate what base 10 means - the expression as the sum of powers of 10.

 

[mfb]

Instead of 372, the book could have used 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 everywhere. This representation works in every base, but I think the problem is obvious. It is easier to write "372" (using the decimal system) even if the topic is about the meaning of this chain of symbols.

 

[Mark44]

Hello.

This is how every number in the decimal system is expressed:

I had understood this topic earlier but as I was revising it today I have become confused somewhat.

I know that for the decimal system, we have 9 digits.

Not quite -- there are 10 digits. Don't forget that 0 is a digit.

I understand this:

- When we use a base between 1-10, we do not need to come up with new digits.

- in the illustration one above, if the base is x, the coefficient a can only be less than x? Am I correct?

Yes. Another base that is commonly used in computer programming, is base-16, or hexadecimal. In this base, there are 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

Another base that is used on the web is base-64, which has 64 digits. I won't list these because I don't know them offhand.

- I am confused as to how to divide numbers and change them to other bases, I seem to have forgotten basic division.

I understand 372 expressed in base 10 is 3*10*10 +7*10 + 2

But what I don't get is this, 372 is already expressed in base 10 when we divide it by 10?

I am really confused. All numbers are intuitively expressed in base 10 before we convert them to other bases..but division by the same base in the same form seems to give the same number.

?
If you divide 372 by 10, you get 37 plus a remainder of 2. That's not the same as 372.
Maybe what you're thinking is how you can "peel off" the digits by dividing by the base.
372 / 10 is 37 with remainder 2
37 /10 is 3 with remainder 7
3/10 is 0 with remainder 3
The remainders, in reverse order, are 3 ... 7... 2

 

[glappkaeft]

Another base that is used on the web is base-64, which has 64 digits. I won't list these because I don't know them offhand.

A-Z, a-z, 0-9, '+' and '/'.

 

[Mark44]

A-Z, a-z, 0-9, '+' and '/'.

With regard to '+' and '/' as the digits with indexes 62 and 63, not necessarily. See https://en.wikipedia.org/wiki/Base64#Variants_summary_table

 

[Logical Dog]

Instead of 372, the book could have used 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.

With regard to '+' and '/' as the digits with indexes 62 and 63, not necessarily. See https://en.wikipedia.org/wiki/Base64#Variants_summary_table

The point is to illustrate what base 10 means - the expression as the sum of powers of 10.

Ok, thanks.