Positional value

S

Saumyojit

Senior Member
Joined
Jan 21, 2020
Messages
1,032
Each position in a decimal number has a value that is a power of 10 .

341=3*10 ^2 + 4*10^1+1*10^0

Why Lsb has the positional value of 10 ^0 ; the subsequent is 10 ^ 1 ...increasing.

What is the logic behind making the postional value start from 0 to the power of base 10 .

I know base is 10 as it is decimal .

in the link : https://courses.lumenlearning.com/collegesuccess2x48x115/chapter/positional-number-systems/

In the Base and Exponent row why it starts from 0.
 
Because [MATH]10^0 = 1.[/MATH]
Thus [MATH]347 = (3 * 100) + (4 * 10) + (7 * 1) = 3 * 10^2 + 4 * 10^1 + 7 * 10^0.[/MATH]
In fact, for any number [MATH]a \ne 0 \implies a^0 = 1.[/MATH]
 
Wait--what? What is your question? I'm rather confused.

Of course you must know that any number except 0 to the power of 0 is one...does that have a bearing on your problem?
 
I know anything to the power zero is 1 but understand my doubt :
Why the Exponent or index of the right most digit is zero

WHy they have used base 10 ... is it because each digit location has a possibility of filling up by anyone of the 10 digits (0-9)?

suppose a 3 digit no will have _10 _10 _10 everytime it has 10 ^1 possibility .
Then 10 ^0 for unit or 10 ^2 for hundreth place have no meaning
 
Let a (non-zero) and b be two digits. Then a*10^1 will be a two-digit number. Also a*10^2 + b*10^1 will be a three-digit number.

Well to have a one-digit number, the number will be a digit! Any digit times 1 will yield a 1-digit number and if you multiply each digit by 1 you get all the digits. Letting the first position being multiples of 10^0 does just what we want.

Is that the answer you are looking for?


In all bases we have this zero power. For example, a number in base 5 will look like a*5^2 + b*5^1 + c*5^0, where a, b and c come from the set {0,1,2,3,4}
 
Any digit times 1 will yield a 1-digit number and if you multiply each digit by 1 you get all the digits thats ok . Now see


assume a no 54 :
if i place 5 in the second place (tens place) then if i think individually i am placing one 5 only in that place .

but u would say 5 * 10 ^1 =50 that means ten 5's .

Isn't it contradicting
 
You have it backwards (although the result is the same) when you say in 54 that you have (or don't have) ten 5s. It is five tens that you have.

No, when you place a 5 in the 2nd position, by DEFINITION, you ARE saying that you have 5 tens! That is why we have values for each position. Otherwise a number like 54 would be 9 (5+4)! What would 56 be? 5+6 = 11 = 2???? Any number you write, if there is no position value, would end up being a single digit number. Do you see this?
 
I have no idea what you are trying to say.

Are you asking primarily about WHY 10^0 = 1?

Or about why we choose to use base 10?

Or about how digits in different places can have different meanings?

Taking your last line literally, you see to be supposing that numbers have to be written something like the ancient Roman or Egyptian numbers, where for example XXX means 30, because there are literally 3 symbols for 10. Our number use "place value" (positional notation), which means that each digit gives the number of a different thing, namely (from the right) ones, tens, hundreds, ... . Putting a 5 in the tens place means 5 tens, simply because that is how the notation works. In Roman numerals, you literally have some number of some symbol, like 3 tens. In positional notation, the position takes the place of the different symbols, and the digit takes the place of the count. To say 30, you put a 3 in the tens place, to mean three tens, rather than three symbols each meaning ten.

I think you probably have several different questions in mind at once. You will need to ask them one at a time, as much as possible, in order to get a useful answer. So perhaps choose one of the three I listed, and tell me why you are unsure of that.
 
@Dr.Peterson
assume a no 54 :
I said that if i place 5 in the second place (tens place) then if i think it as individually then I am placing one 5 only in that place not 5 tens .

why we are multiplying a number wtih 10 ^ 0 at unit place.Why zero is exponenet and base is 10
 
Do you actually not understand that the 5 in the tens place makes the value 50? Do you think you should write 5555555555 to mean 50?

Do you not understand that 10^0 = 1, or are you asking for proof of that fact? You are not being clear.

One way to clarify a "why" question is to propose an alternative. What do you think these should be instead?
 
thats is what i am asking why 5 is getting multiplied with 10 ^ 1 .

why at units place we are mutliplying the no with 10 ^0.

give me proof how this exponent n-1 where n is no of digits to the power of 10 has evolved or came.

when we are expanding any digit no(24) how we are expressing them into 2*10^1+4*10^0
how 10 ^1 and 10 ^0 came ...how the counting happens in the backend

why base 10 and why exponent n-1....
 
Saumyojit said:
why base 10 and why exponent n-1....
Click to expand...
Why base 10? That is the easiest question to answers: humanoids have ten digits.
The next part takes considerably more to answer. Consider the five digit number, \(N=84526\).
The first five natural numbers are \(0,1,2,3,4\). [now I know math-ed types do not like that to which I say get-over-it]
Now build \(N\) in terms of powers of \(10\). Now the unit digit of \(N\) is \(6\) and \(6=6\cdot10^0\) Note the exponent is the first natural number.
Now the tens digit of \(N\) is \(2\) and \(20=2\cdot10^1\) Note the exponent is the second natural number.
The hundreds digit of \(N\) is \(5\) and \(500=5\cdot10^2\) Note the exponent is the third natural number.
Lets cut to the chase observe that \(N=6\cdot 10^0+2\cdot 10^1+5\cdot 10^2+4\cdot 10^3+8\cdot 10^4\)
To completely understand this notation think about \(9\cdot 10^7=90000000\). That is a \(9\) followed by seven zeros.
So \(4856=4000+800+50+6\). YOU count the zeros in each term of the sum. Are they \(3,2,1,0~?\)
 
u said that :
The first five natural numbers are 0,1,2,3,4.

Why should i take first five ones only? i could have taken first 4 also.

Why should i build N in terms of powers of 10.? that is the whole point i am asking in this doubt. Why we are taking zero to power of 10 and then multiplying with the digit of each place.

u said the logic behind it is: YOU count the zeros in each term of the sum .
why should i count the zeroes of each sum.!
 
You are asking about NUMERALS rather than numbers. Theere is no logical reason that the most common system of numeration is based on combining ten symbols; like all systems of coding, it is ultimately arbitrary. Hexadecimal numerals are based on combining sixteen symbols. Binary numerals are based on combining two symbols. Octal numerals are based on combining eight symbols. The ten symbols used in the decimal system are also arbitrary. In that system

0 means zero
1 means one
2 means two
3 means three
4 means four
5 means five
6 means six
7 means seven
8 means eight
9 means nine.

If we were talking about hexadecimal numerals, we would use sixteen symbols, and

D would mean fourteen.

Numerals are concise ways to represent numbers.

In the decimal system, to represent the first ten non-negative integers, we use exactly one of the symbols. To represent the first hundred non-negative integers, we use one or two of the symbols. In the octal system, representing nine requires the use of two symbols, and representing seventy requires the use of three symbols.

In the decimal system, 594 represents the number that equals the sum

[MATH]500 + 90 + 4.[/MATH]
But that equals a sum of products, namely

[MATH]5 * 100 + 9 * 10 + 4 *1.[/MATH]
And the second term in each product is a power of 10, so

[MATH]594 = 500 + 90 + 4 = 5 * 100 + 9 * 10 + 4 * 1 = 5 * 10^2 + 9 + 10^1 + 4 * 10^0.[/MATH]
Different ways to represent the same number.

No more mysterious than that the French say "chien" whereas the English say "dog." Representations are arbitrary.
 
Saumyojit said:
Why should i take first five ones only? i could have taken first 4 also.

Why should i build N in terms of powers of 10.? that is the whole point i am asking in this doubt. Why we are taking zero to power of 10 and then multiplying with the digit of each place.
Click to expand...
Here is one way to understand why we use positional notation, and why it works as it does.

I introduced the concept to my son using beans, cups, and trays. We put 10 beans in a cup, and 10 cups on a tray. So in counting, say, 247 beans, we could start filling a cup, and when it reached 10, put it on a tray and start a new cup. So every 10 beans made 1 cup. After the first 100 beans, we would have 10 full cups on a full tray, and would get out a new tray. Eventually, we would have 2 full trays, 4 extra full cups, and 7 more beans: 247. (Of course, we didn't actually do all this more than a couple times before finding quicker ways to count! Then we started adding numbers by combining cups.)

Do a base-ten numeral just states the counts of trays, cups, and beans (hundreds, tens, and ones) in a very compact, efficient way.

Why powers of ten? Because that's as far as you can count on your fingers, and because it's not too small and not too large. The number 34 can be shown as 3 pairs of hands and 4 more fingers. Even before base-ten positional notation was invented, names in many languages included special words for ten and hundred; positional notation just made it easier to write large numbers, by leaving out the words or special symbols for them.

Why use digits only up to 9? Because the next number is a "full cup", and we can start counting tens. We never need more than 9 of anything in this system. Again, to do otherwise would be more complicated and unnecessary.

Now, even the Romans with their complicated symbolic system understood place value: they used something like an abacus, where stones in one place (like beads on one wire) counted ones, those in the next counted tens, and so on. They just didn't get the idea that the value of a written symbol could be implied by its position, just as the value of a stone or bead could be.

I imagine it took Europeans some time to get used to the idea when place value was introduced (from India by way of the Middle East), but they soon saw its value, even if they didn't quite see why. And, of course, it was only later that it was commonly expressed in terms of powers of ten, as exponents hadn't quite been invented yet, and most people wouldn't understand 10^0. But that was all hidden in the concept.
 
I shudder while thinking the day the student is introduced to binary or hexadecimal or octal numbers!!:ROFLMAO::ROFLMAO::ROFLMAO:
 
@Dr.Peterson Why use digits only up to 9? Because the next number is a "full cup", and we can start counting tens. We never need more than 9 of anything in this system.
this line means that when i am facing 9 that means i am ending with units system right...but how we are placing zero in place of 9 and to the right 1 . How we are recounting from 9...?
 
Last edited:
You must log in or register to reply here.
Top