What major topics are there in arithmetic and what order to learn them?

[mark2142]

TL;DR Summary: What topics to cover to safely say I know arithmetic ?

I am learning arithmetic from Indian NCERT textbook. Currently I have finished addition ,substraction of 2 digit numbers and divisions, multiplication of 1 digit numbers. I am moving pretty slowly. Can someone tell me what topics to cover first to build a framework and then go on in detail. I want to learn fast. It has taken me a year now learning arithmetic. I want to speed up. Thanks for the help in advance. (I also have a book called arithmetic for the practical man by Thompson.)

 

[fresh_42]

I could be wrong since your description lacks details, so I have to make guesses. This sounds to me as if you were currently studying something like taught in this book:
https://openstax.org/details/books/prealgebra-2e

You could go on from there to the other (free pdf) books on that webpage.

 

[jedishrfu]

Here’s a youtube video called the Map of Mathematics

 

[mark2142]

I could be wrong since your description lacks details, so I have to make guesses. This sounds to me as if you were currently studying something like taught in this book:
https://openstax.org/details/books/prealgebra-2e

You could go on from there to the other (free pdf) books on that webpage.

1. How did indians come up with such positional system ? Even though its simple but Looks mysterious.

2. Why 400 + 20 +3 is written as 423 ? Why have we omitted zeros ?

3. Why four hundred is written as 400 and not as 4 hundred or 4. Similarly why twenty is written as 20 and not as 2 ten or 2 ?

4. What is place value notation ? Is it different from place value ?

 

[WWGD]

1. How did indians come up with such positional system ? Even though its simple but Looks mysterious.

2. Why 400 + 20 +3 is written as 423 ? Why have we omitted zeros ?

3. Why four hundred is written as 400 and not as 4 hundred or 4. Similarly why twenty is written as 20 and not as 2 ten or 2 ?

4. What is place value notation ? Is it different from place value ?

It seems you're looking more for history of Mathematics and Arithmetic than Arithmetic itself.

 

[jedishrfu]

Jan Gullberg's book:

https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X?tag=pfamazon01-20

It's a good place to start, part historical and part tutorial up to first-year college math.

I don't think it has Eskimo math:

https://en.wikipedia.org/wiki/Kaktovik_numerals

Invented by school children in 1994, to go along with their cultural traditions and use of base-20 math in their native language.

 

[fresh_42]

1. How did indians come up with such positional system ? Even though its simple but Looks mysterious.

2. Why 400 + 20 +3 is written as 423 ? Why have we omitted zeros ?

3. Why four hundred is written as 400 and not as 4 hundred or 4. Similarly why twenty is written as 20 and not as 2 ten or 2 ?

4. What is place value notation ? Is it different from place value ?

These are interesting questions and not easy to answer. As far as I know, the Indians were the first culture to give zero a symbol that ended up in our 0. The Babylonians, later, also introduced a symbol for an empty space. Western history writing basically starts with the Sumerians. I had a hard time searching for Indian sources, and it was even harder to figure out what had been migrated from India to Sumer or Babylon. They are not that far away from one another, so there might have been a cultural exchange along trading paths.

I like to consider the finding of zero as the beginning of mathematics: Someone decided to count what wasn’t there! Just brilliant! However, the truth is as often less glamorous. Babylonian accountants needed a placeholder for an empty space for the number system they used in their books. The digits zero to nine have been first introduced in India. In Sanskrit, zero stands for emptiness, or nothingness.

Source: https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/

 

[symbolipoint]

2. Why 400 + 20 +3 is written as 423 ? Why have we omitted zeros ?

3. Why four hundred is written as 400 and not as 4 hundred or 4. Similarly why twenty is written as 20 and not as 2 ten or 2 ?

I cannot say why from History, but the meaning of Place Value starts very early in the instruction schedule. Right this moment I am not prepared to try explaining Place Value.

I might only say this much: Number 423, the "4" is counting the hundreds; the "2" counts the tens; and the "3" counts the ones. And if you may want to ask, "So what about the thousands?", we must say, the "0" counts the thousands. So you may as, "then why is this 0 for the thousands not shown?"

 

[WWGD]

For contrast, the OP @mark2142 , you may want to consider the likes of Roman Numerals, which are not positional. Try to multiply CLXI by MVII.

 

[CrysPhys]

I might only say this much: Number 423, the "4" is counting the hundreds; the "2" counts the tens; and the "3" counts the ones. And if you may want to ask, "So what about the thousands?", we must say, the "0" counts the thousands. So you may as, "then why is this 0 for the thousands not shown?"

<<Emphasis added.>> For the same reason that you don't show 0 for ten thousands, hundred thousands, ... ad infinitum when not needed. But you would show 0 for ten thousands when needed; e.g., 60423.

 

[fresh_42]

For contrast, the OP @mark2142 , you may want to consider the likes of Roman Numerals, which are not positional. Try to multiply CLXI by MVII.

That threw us 1,500 years behind.

 

[WWGD]

That threw us 1,500 years behind.

Is there a more modern one that we can use as example?

 

[fresh_42]

Is there a more modern one that we can use as example?

I don't think so. The Babylonians needed the zero for their bookkeeping, and in the Middle Ages, it was the traders who switched to the Arabic (aka Indian) number system for obvious reasons, e.g., counterfeit security. The economy is a hard impetus for new trends. So why should anyone go back?

The strangest things are p-adic numbers, which have dots on the left:
$$
\ldots + 3\cdot 5^3 + 3\cdot 5^1+3\cdot 5^{-1}=\ldots 3030_5.3=\overline{30}_5.3=-\dfrac{1}{40}_{dec}
$$

 

[WWGD]

I don't think so. The Babylonians needed the zero for their bookkeeping, and in the Middle Ages, it was the traders who switched to the Arabic (aka Indian) number system for obvious reasons, e.g., counterfeit security. The economy is a hard impetus for new trends. So why should anyone go back?

The strangest things are p-adic numbers, which have dots on the left:
$$
\ldots + 3\cdot 5^3 + 3\cdot 5^1+3\cdot 5^{-1}=\ldots 3030_5.3=\overline{30}_5.3=-\dfrac{1}{40}_{dec}
$$

Maybe to illustrate the ineffectiveness of such system? As advanced in many ways as they were, they were surprisingly backwards in their number system. Yet somehow they produced some advanced engineering and architectural works that were ahead of their times.

 

[fresh_42]

As advanced in many ways as they were, they were surprisingly backwards in their number system. Yet somehow they produced some advanced engineering and architectural works that were ahead of their times.

But they didn't calculate. They relied on experiences and (Greek) geometry.

Even Roman numerals are a place value system, although quite inconvenient.

 

[symbolipoint]

<<Emphasis added.>> For the same reason that you don't show 0 for ten thousands, hundred thousands, ... ad infinitum when not needed. But you would show 0 for ten thousands when needed; e.g., 60423.

Yes. I did intuitively understand that as I was asking. I just was not interested in trying to explain. You did a more efficient job of it.

 

[gmax137]

The strangest things are p-adic numbers, which have dots on the left:

I watched a couple of YouTubes on p-adics, blew my mind.

 

[WWGD]

I watched a couple of YouTubes on p-adics, blew my mind.

They're another way , i.e., besides the standard Real numbers, of metrically completing the Rationals, i.e., plugging the holes that allow $x^2=2$ , to have no Rational solution.

 

[mark2142]

These are interesting questions and not easy to answer. As far as I know, the Indians were the first culture to give zero a symbol that ended up in our 0. The Babylonians, later, also introduced a symbol for an empty space. Western history writing basically starts with the Sumerians. I had a hard time searching for Indian sources, and it was even harder to figure out what had been migrated from India to Sumer or Babylon. They are not that far away from one another, so there might have been a cultural exchange along trading paths.

I am just curious to know what was going in the mind of brahmagupta or aryabhatta that they came up with such a system that is efficient in representing numbers unlike roman numerals. Large numbers can be written with indian system but it would be difficult to do it in roman. It would create long array of symbols which would be hard to remember and write i guess.

I might only say this much: Number 423, the "4" is counting the hundreds; the "2" counts the tens; and the "3" counts the ones. And if you may want to ask, "So what about the thousands?", we must say, the "0" counts the thousands. So you may as, "then why is this 0 for the thousands not shown?"

Thanks for confirming. I was thinking like that but not sure. I guess we can write 0423 or 00423 but no need to. It would be same as 423.

It seems you're looking more for history of Mathematics and Arithmetic than Arithmetic itself.

I don’t want that. Haha…

 

[mark2142]

Try to multiply CLXI by MVII.

In Indian 161 by 1007= 1,62,127. In Roman it would be CCC…(MVll times)LLL…(MVll times)XXX…(MVll times)lll…(MVll times).

 

[mark2142]

I cannot say why from History, but the meaning of Place Value starts very early in the instruction schedule. Right this moment I am not prepared to try explaining Place Value.

I might only say this much: Number 423, the "4" is counting the hundreds; the "2" counts the tens; and the "3" counts the ones. And if you may want to ask, "So what about the thousands?", we must say, the "0" counts the thousands. So you may as, "then why is this 0 for the thousands not shown?"

I think 423 is actually 4 hundreds 2 tens 3 ones but for convenience we don’t write hundreds, tens, ones. It’s easy if we define the concept of place value. For example 3rd place is hundredth place, 2nd place is tens place and 1st place is one’s place and just write 423 directly in short.

1. One last question : 4* 10^2 + 2*10^1 + 3*10^0 =1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 =7*60^1 + 3*60^0. All are equal to number 423 in base 10. Is place value system just a way of representing numbers in terms of some particular number like 10, 7, 60. We can choose a number, try to calculate the sum and then write only the coefficients for example 4,2,3 in base 10. 1,1,4,3 in base 7. 7,3 in base 60. So number becomes 423 in base 10. 1143 in base 7. 73 in base 60 ??

(I don’t know how to change from one base to another)

 

[WWGD]

In Indian 161 by 1007= 1,62,127. In Roman it would be CCC…(MVll times)LLL…(MVll times)XXX…(MVll times)lll…(MVll times).

Yes, I never figured out your "Crore" system either.

 

[symbolipoint]

mark2142,
We found or designed base-two because of how simple it is; I do not mean "easy"; I mean "simple".
I imagine base-ten was designed because we have ten fingers; or we have ten toes; but seems fingers are more convenient. Then while counting or computing, we do not have enough fingers to show what we want. .... and on, and on, and on...

 

[mark2142]

mark2142,
We found or designed base-two because of how simple it is; I do not mean "easy"; I mean "simple".
I imagine base-ten was designed because we have ten fingers; or we have ten toes; but seems fingers are more convenient. Then while counting or computing, we do not have enough fingers to show what we want. .... and on, and on, and on...

Yeah.

1. One last question : 4* 10^2 + 2*10^1 + 3*10^0 =1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 =7*60^1 + 3*60^0. All are equal to number 423 in base 10. Is place value system just a way of representing numbers in terms of some particular number like 10, 7, 60. We can choose a number, try to calculate the sum and then write only the coefficients for example 4,2,3 in base 10. 1,1,4,3 in base 7. 7,3 in base 60. So number becomes 423 in base 10. 1143 in base 7. 73 in base 60 ??

Can you shed some light on this ?

 

[mark2142]

Yes, I never figured out your "Crore" system either.

1. It’s just the difference in placement of commas as much as I know.

2.It would be huge array of symbols in Roman.

 

[symbolipoint]

Can you shed some light on this ?

Not really. Some of your expression or statement seems to use base-seven; so too hard for me to think through this. Maybe someone else can explain. And later, base-sixty? Seems too long to do anything with it. Somebody else!

 

[mark2142]

Not really. Some of your expression or statement seems to use base-seven; so too hard for me to think through this. Maybe someone else can explain. And later, base-sixty? Seems too long to do anything with it. Somebody else!

But that is what i am asking. A power has two parts. Base and exponent.

1. 4* 10^2 + 2*10^1 + 3*10^0 = 423
2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143
3. 7*60^1 + 3*60^0 = 73

All three expressions are equal in quantity. I am just trying to say that what we call place value system is we are writing the same number by just changing the base of power in three expressions. Then writing the coefficients only in the expressions namely 4,2,3 in first expression forming the number 423, 1,1,4,3 in second expression forming the number 1143 which is again 423 in base 10 and 7,3 in third expression forming the number 73 which is also 423 in base 10 and 1143 in base 7. Every time the base is changed in expression without changing the value of the number. This is what place value system is in essence. Change the base of power in expressions and the digits will change automatically but it will have no effect on the value of the number.