Understanding decimal number system

Question

I am just starting with number systems and I am reading decimal system right now, I got these 2 points which I am unable to understand, any explanation would be great for both of them

The value of the base in all positional number systems suggests the following characteristics.

1. The value of the base determines the total number of different symbols or digits available in the number system. The first of these choices is always zero.

2. The maximum value of a single digit is always equal to one less than the value of the base.

Answer 1

The easiest example would be base $10$ which is the number system you grew up with.

What does $56074$ mean. It means

$5\times 10^4 + 6\times 10^3 + 0\times 10^2+ 7\times 10 + 4\times 1=$

$50000 + 6000 + 0 + 70 + 4 = 5674$.

In this our number system is based on a base, $b$. In the case $b=10$.

So

1. The value of the base determines the total number of different symbols or digits available in the number system. The first of these choices is always zero.

This says our choice of base, (In our case $10$) determines how many digits we have. (In our case we have $10$ digits. ) Furthermore our first digit is $0$. (An in our case our firs digit is $0$; and we have $10$ digits total: They are $0,1,2,3,4,5,6,7,8,$ and $9$.)

2) The maximum value of a single digit is always equal to one less than the value of the base.

SO in our case our largest digit is $9$. And $9$ is one less than $10$. (Is that a coincidence? Or was that on purpose?... It was on purpose.)

So in base $10$, we have $10$ digits, our first digit is $0$, and our largest digit is $10-1 = 9$.

....

Suppose we want to do a number system not based on $10$ but on,say, .... $4$.

Our system base on $4$ will

1) Have $4$ digits. The smallest is $0$.

and 2) the largest is $4-1 = 3$.

So we have $4$ digits they are $0,1,2,3$.

And so we count: Our first three numbers are $1_4,2_4,3_4$ but no we've run out of digits so we go on to $10_4,11_4,12_4,13_4$ but now we've run out so we go to $20_4,21_4,22_4,23_4$ etc.

So a number in a base $4$ system will be represented as:

$12023_4= 1\times 4^4 + 2\times 4^3 + 0\times 4^2 + 2\times 4 + 3\times 1$

Just as every number can be represented by a string of digits (each digit $0$ to $9$) with each digit representing a number of powers of $10$. (And even if you never thought about it, this is why our number system works). We could just as well represent every number by a string of digits, each digit, $0$ to $3$, with each digit representing a number of powers of $4$. After all theres nothing magical, other than our ability to count on our fingers, about having ten digits.

(And you can be sure that if we had $8$ or $12$ fingers we would have developed a different number system. Have you thought about what it would have been?)

Answer 2

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). For base 10 there can be 10 symbols or digits i.e. 0 to 9 Here 9 is the the maximum value. For example- 209 can be written as-

2 x 10^2=200

+0 x 10^1=0

+9 x 10^0=9

             =209

10 x 10^x can be 10^(x+1)

Answer 3

It's possible to break the first rule as that's more of what the radix does. It is the usual case that the radix is equal tot he base, but it need not be. The second point is also only valid under this normal case.