1 Why we count the way we do

Have you ever wondered why we can write down any numbers we please using only ten different symbols 0 to 9? Compare the decimal number system to the Roman number system disaster.

We count the way we do, because numbers are based on the ideas of position and position weight.

Knowing how to count is a prerequisite to any mathematical activity. Counting is ground zero. A major goal is to be able to write down a number, which represents any value you choose, and, at the same time to know why numbers are written the way they are. Knowing why means you understand the ideas contributing to the rules of the number system we use every day.

Learning to count requires knowing the symbols for numbers as well as the names of numbers. At first, mathematics is difficult because there are all of those new symbols and words to learn before you can really get going.

Any number represents some quantity of units. Know that the unit can be apples, cars, or whatever. The house is 27 feet wide. No, no the house is 9 yards wide. Ultimately we choose one, one of anything, (whose symbol is 1) as our unit. Thus any number represents some quantity of ones. The symbol 5 represents the sum of five ones. Mathematics benefits immensely from abstractions such as 1, without committing to 1 whatever (feet, yards, cars, …).

If you select any number, the next number is found by adding 1. In the abstract language of algebra, if letter n is shorthand for the word number and n represents any number, then the next number is n+1.

Note: Sooner or later we learn to count from zero to infinity. Infinity is an alias for a number that is as large as you please.

1.1 Zero to Nine: 0 to 9

Over the centuries people have agreed on names for numbers. There is, however, a very big problem with names, which are just words. The very big problem is that we cannot use words to do arithmetic, because that would be tedious and probably impossible.

We can do arithmetic if we use symbols in combinations. A study of the history of number reveals that the world has agreed on (arbitrary) names and symbols for quantities ranging from zero to nine.

A number line is a geometrical representation of number. The very important number line is a graphic display of numbers. The question mark on the line is discussed in upcoming paragraphs.

The number line is constructed by marking off equal lengths along the line. Each mark on the number line is assigned a number. Assign 0 to any mark. Next, assign 1 to the first mark to the right of zero. Then the distance from 0 to 1 represents 1 unit of length. Subsequent marks to the right add 1 unit to the distance. Label subsequent marks 2, 3, 4, and up to 9.

1.2 Counting past 9

When we try to count units beyond 9 we run out of number symbols, because the world decided not to introduce new symbols. The world not want to repeat the disaster of the Roman number system (such as XLVII or 47).

When we count units, all we can do with units is to count from 0 to 9 over and over again. We recycle through 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, etc. Each time we reach 9 we have counted an additional 9+1 units.

This number 9+1 needs a name, and just how do we represent 9+1? Well, the number 9+1 has to be one whatever, and, if we continue counting. the next 9+1 makes it two whatevers. The whatevers have to be counted by another digit taken from 0 to 9, because no new symbols are allowed.

Another digit would allow us to add 1 to it each time the unit’s count recycled from 9 back to 0 (another whatever). This other digit records the number of 9+1’s. This digit needs a name. Someone must have said abracadabra and arbitrarily named the next number 9+1 ten, and so the other digit is the ten’s digit. No new symbols means using only 0 to 9 as the ten’s digit, All digits are 0 when we start counting.

As we count, units increment from 0, and each time the units recycle from 9 back to 0 to start over, the ten’s digit is incremented by 1 as shown here. The count shown here is from 00 to 42 if you join the vertically aligned digits.

The situation, however, is unsatisfactory. Here is the unit’s digit and over there is the ten’s digit. What is a useful way to associate the unit and ten digits, so that we can calculate with numbers greater than 9?

After a long while, there was agreement to write numbers as if they were words. The agreement was to write the digits in a sequence with the highest value digit first as in 10, which one could read as 1 ten and 0 units. In this way, ten became the two digit number 10 (one, zero).

Now we know the question mark ? on the number line is a 10

We count from 0 to 9, 0 to 9, etc. Each time we count through 0, we increment the ten’s digit as shown above. We continue counting until we reach 99 units, when we are faced with the question, what to do with 99+1?

We observe that 99+1 represents 9 tens + 9 units + 1 unit. We choose to convert the 9 units + 1 unit to 1 ten so that we can say 99+1 represents 9+1 or 10 tens. Consistent thinking produces yet another digit, a third digit, that counts 9+1 tens. Since names are arbitrary we say abracadabra again and the name is hundred’s digit.

This means when we count tens from 0 to 9, 0 to 9, etc., we increment the hundred’s digit each time we count tens through 9 to 0. An important equivalent statement is we increment the hundred’s digit each time we count tens/units up to 99, and recycle through 99 to 00. When tens and units make 99, and 1 is added, there is a roll over into hundreds. The count here is from 000 to 420.

The first time we count tens/units through 99 to 00, or 099 to 000, we increment the hundred’s digit past 0. Consequently we write 99+1 in word format as 100 (one, zero, zero).

Can you guess what’s next? We continue to count and soon we reach 999. The next number is 999+1. The name is thousand, and we write it in word format as 1000 (one, zero, zero, zero). When units, tens, and hundreds equal 9, then the next +1 causes a roll over into the next thousand.

In this way, counting units forces a counting of tens. And, in the same way, counting tens and units forces a counting of hundreds. And, again in the same way, counting hundreds and tens and units forces a counting of thousands from 0000 to 4200 here. We are on our way to infinity.

1.3 Zero to Infinity

There is always a greater number. As mentioned above to any number n you add 1 to make the next number n+1.

For example, increase the number 9999 by one unit. What do you get?

Nine thousand, nine hundred, ninety, nine, plus 1 is nine thousands, plus nine hundreds, plus nine tens, plus ten ones so you get ten thousands.

We write nine zero zero zero, 9 0 0 0 as 9000 for nine thousands. And then we write ten zero zero zero, 10 0 0 0, as 10000 for ten thousands? This is consistent with 10 0 0 or 1000, 10 0 or 100, and 1 0 or 10. Now we have a five digit number.

Add one to get ten thousand one, 10001.

Add one to 100 000 000 to get 100 000 001 (To be clear we have separated the digits into groups of three.)

Add one to 345 543 672 879 to get 345 543 672 880.

And on to infinity.

The genius of the decimal number system is that we only need the symbols 0 to 9, and the concept of position, to write any number we please, such as the numbers:

Then the concept of position weight gives each number a unique value.

1.4 The Ideas of Position and Position Weight

The four digit number 1876 has four digits occupying four positions. Each position can only contain one digit, which is taken from the list of one digit numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The four positions are filled by 1, 8, 7, and 6 respectively to create the number 1876. In other words, numbers greater than nine must have two or more digits.

We are not limited to four positions. We can have an unlimited number of positions. The rules are that any digit from 0 to 9 can be written in any position, and there can be as many positions as we desire. This is why we can write down any numbers we please such as these using only ten different symbols. We consider this to be a remarkable number system.

The positions are assigned numbers starting with 0 at the right hand digit. The digits of 1876 are in positions 3, 2, 1, 0. The significance of defining position zero as zero instead of one becomes very clear when you study decimals. There you learn about the idea of powers of ten and that 1010 to the zero power, 100, equals 1.

Now what? We need a new idea. In fact it was a great idea that someone unknown to us revealed a long time ago. The idea is assigning different weights for different positions in a number. (We used 1, 10, 100, …. weights when we counted past 9.)

When we count past 9 we have accumulated tens, hundreds, and so forth, which we report by incrementing the digit to the left when all digits to the right are 9. Each digit in a number is treated in the same way. This display shows hundreds incrementing by 1 as units and tens pass through 99 to 00.

This is why giving each digit position a number and a weight turns out to be the key steps in the process leading to larger numbers.

Base ten What are the words describing the values of these numbers?

The weights of the positions have names. These are the base ten names. Base ten, because the weight increases ten times when we move one digit left (e.g. you add a zero to 100 to produce 1000). We include spaces for clarity when we write large numbers.

The number 2705 Now we can say the number 2705 consists of two thousands, seven hundreds, zero tens, and five ones. We read the number 2705 as two thousand, seven hundred, five. Zero digits are not voiced

Position In the number 116210 digit 6 is in position 3.

Weight and value The weight of position 3 is 1000, and a digit 6 in position 3 contributes the value 6000 to the number 116210.

We distinguish position weight from value a digit contributes. This is why we say the weight of position 3 is 1000, and the value of digit 6 in position 3 is 6×1000=6000.

1.5 Review

The Basic Ideas of Counting Start counting from zero. Count up from 0 by adding one to get the next number: zero, one, two, three, four, five, six, seven, eight, and nine. When 9 is reached adding 1 converts the 9 to 0 and a 1 is added to the left of the 0. We say 09 rolled over to become 10. This happens each time a nine is reached: e.g. 9 to 10, 39 to 40, 519 to 520, 999 to 1000. This is why we only need the ten symbols 0 to 9 to write out any number.

Position and position weight A position number is assigned to each digit in a number. The position numbers match the digits in the contrived number 9876543210. A number’s right hand digit is in position zero. The digit 3 of the number 5603 is in position zero.

The position weight of position 2 is 100 so that the value of the 6 in position 2 in the number 5603 is 6×100 or 600 units. The position weight of position 3 is 1000, and so the digit 5 represents 5000 units.

Note that the number of zeros in the position weight matches the position number: For example the weight of position 9 is 1 000 000 000.

The digit in any position tells you how many position weights contribute to the number: for example the 6 in 456071 contributes six position 3 weights of 1000, and the 0 contributes zero position 2 weights of 100. This is why placing zeros to the left of a number (000456071) does not change the number’s value.

One digit per position One reason this number system works is that each position is occupied by only one digit.

Only 0 to 9 in any number As we have said any digit from 0 to 9 can be written in any position, and there can be as many positions as we desire. This why we can write down any numbers we please such as these using only ten different symbols. We consider this to be a remarkable number system.

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