The Development of Numbers The Development of Numbers next Taking the Fear out of Math © Math As A Second Language All Rights Reserved Hieroglyphics Tally Marks Roman Numerals Sand Reckoner

Gives Way to next © Math As A Second Language All Rights Reserved The Sand Reckoner The Sand Reckoner Place Value Place Value

Inventing Our Modern Place Value System next © Math As A Second Language All Rights Reserved Although the “nouns” on the sand reckoner all looked alike, they could still be distinguishable from one another by their relative position. This property of the sand reckoner is essential in understanding the next stage of abstraction in the development of our number system, known as place value.

The development of our place value system for representing numbers ranks as one of the great inventions of the human mind. next © Math As A Second Language All Rights Reserved Thus, again based on our having ten fingers, symbols (numerals) called digits were introduced to represent the numbers we know as one, two three, four five, six, seven, eight, and nine, and are denoted by the symbols 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. next

As in the sand reckoner, the digits were placed next to one another in a row with the digits playing the role of the stones. © Math As A Second Language All Rights Reserved For example, in the new system the number represented on the sand reckoner as… next would now appear as… 4213

© Math As A Second Language All Rights Reserved In this new invention, while the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 serve as the adjectives, the nouns (that is, the denominations) for the first time were invisible. The denominations could only be determined by where a digit was placed; hence, the name place value. Key Point

next Inventions are not made “in a vacuum”. When we talk about the ideas that go into the concept of place value, notice how many of its features had been invented earlier. © Math As A Second Language All Rights Reserved “If we seem to see further than others, it is because we stand on the shoulders of giants.” Notes “Sir Isaac Newton

next ► Place value is abstract, but so are tally marks. © Math As A Second Language All Rights Reserved Notes ► Place value is based on “trading in” by tens, but so are Roman (and Egyptian) numerals. ► Place value is based on each denomination being worth ten of the denomination immediately to its right), but so is the sand reckoner. next

► What made place value different is that for the first time one could not recognize a denomination until one saw what place the digit was in. © Math As A Second Language All Rights Reserved Notes

next Consider the following four numbers as represented on a sand reckoner… “The fact that in place value the noun that a digit represented could be determined only by the position of the digit led to a serious problem. © Math As A Second Language All Rights Reserved

next Even though in each picture there are 2 stones followed by 3 stones, we can see that each of these numbers is different by looking at the lines on which the stones are placed. © Math As A Second Language All Rights Reserved

next But if we use the place value representation of these numbers in which the lines (or denominations, or nouns) are invisible, these numbers have the same sequence of digits and can only be distinguished by the spacing between the digits. © Math As A Second Language All Rights Reserved

next 2 3 © Math As A Second Language All Rights Reserved Obviously, when all we write is 2 3 or even 23, it is impossible to tell what denominations are being modified by 2 and 3. next

For this reason, we introduce a place holder to make completely clear which noun is being modified by which adjective, even though the nouns still remain invisible. © Math As A Second Language All Rights Reserved The digit, or symbol, that one uses for this purpose is 0, and the name we give this symbol is zero. 1 note 1 It is easy to confuse 0 with “nothing”. Just as 3 is a digit that tells us that we have three of a particular denomination, 0 is a digit that tells us that we have none of a particular denomination. For example, the place holder, 0, allows us to distinguish the difference between 40 and 400 in the sense that we can see that in 40 the 4 modifies tens while in 400 it modifies hundreds. next

© Math As A Second Language All Rights Reserved 2 3 In any case, with the introduction of 0 as the place holder, the above four numbers have their familiar place value appearance 23, 203, 2003, and 2300.

next ► The main point is that we need 0’s to distinguish 2,300 from 2,003 because the nouns are named solely by the position of the digits. © Math As A Second Language All Rights Reserved Notes ► This problem didn’t exist with the use of Roman numerals because there is never a danger of us confusing MMCCC with MMIII (because CCC doesn’t look like III). next

In summary, the invention of zero as a place holder is one of the ranking achievements of the human mind. © Math As A Second Language All Rights Reserved Its importance for the development of mathematics, science and technology can not be overstated, especially in terms of how it, in combination with our adjective/noun theme, simplifies the study of arithmetic. next

The advent of place value made it easier to represent (as well as comprehend) much greater numbers, and as science and technology improved, numbers became sufficiently large that even by trading in by tens, the process eventually became tedious. © Math As A Second Language All Rights Reserved For example, think of how tedious it would be to try to read a number such as… next

next In other words, it is just as difficult to keep track of the number of digits in as it is to keep track of the number of tally marks in the number named by | | | | | | | | | | | | | | | | |. © Math As A Second Language All Rights Reserved Thus, the next evolutionary step in developing our number system was the subtle discovery that we could just as easily “trade in” by thousands as we could by tens. next

By way of analogy, when we see the word “cat” we recognize it without having to “sound out” each letter. Similarly, when we see the place value numeral 647, we recognize it immediately as representing “six hundred forty seven,” without sounding out the denominations. © Math As A Second Language All Rights Reserved We do not say, “Let’s see, the 6 is in the hundreds place, the 4 is in the tens place, and the 7 is in the ones place”. next

Thus, rather than just using the denominations “ones”, “tens” and “hundreds” and the adjectives 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to modify them, more comprehensive denominations such as “thousands”, “millions”, “billions”, “ trillions”, “quadrillions”, were introduced, and one thousand adjectives were used to modify them. © Math As A Second Language All Rights Reserved These one thousand adjectives are the numbers we write as 0 (written as 000), 1 (written as 001), 2 (written as 002), 3 (written as 003), all the way up to 999. next

In this way, the hard to read numeral becomes the easier to interpret numeral… © Math As A Second Language All Rights Reserved which we read as… 8 quadrillions, 945 trillions, 600 billions, 893 millions, 49 thousands, 875“units” next hundred ten one h t o quadrillionstrillionsbillionsmillionsthousandsunits next

As long as the denominations are visible there is no need to write the 0’s. We could have written… © Math As A Second Language All Rights Reserved The 0’s are needed only when the denominations are not visible. next hundred ten one h t o quadrillionstrillionsbillionsmillionsthousandsunits Notes

next In this case, we use commas to separate the denominations. Starting with the digit furthest to the right, we count from right to left in groups of three digits (including 0’s). In this methodology, we would write the above number as… © Math As A Second Language All Rights Reserved 8,945,600,893,049,875. next hundred ten one h t o quadrillionstrillionsbillionsmillionsthousandsunits Notes

next Notice that each large denomination is further modified by the number of ones, tens and hundreds; e.g., hundred millions, or ten quadrillions. © Math As A Second Language All Rights Reserved 8,945,600,893,049,875 next Notes For example, in the above number, the ‘6’ is in the hundreds place and the hundreds place is modifying “billions”. Hence, we see that ‘6’ is modifying a hundred billion.

next No matter how large the noun (i.e., denomination), we never have to deal with an adjective that is greater than 999. © Math As A Second Language All Rights Reserved next Notes The word million is associated with the third large denomination from the right, and as such it may be thought of as a thousand thousands (in much the same way that we can look at 200 as being 200 ones, 20 tens or 2 hundreds).

next The prefixes “bi”, “tri”, “quad”, “quint”, “sept”, “oct” etc., give rise to the denominations billion, trillion, quadrillion, quintillion, sextillion, septillion, and octillion etc. In terms of our adjective/noun theme, the “units” are the name of the noun we are describing. © Math As A Second Language All Rights Reserved next Notes Thus, if we are talking about grains of sand, the number would be… 8,945,600,893,049,875 grains of sand

next © Math As A Second Language All Rights Reserved next Notes However, even with this advanced system of enumeration, it would be extremely cumbersome to name a number such as… 10,000,000,000,000,000,000,000,000,000,000

next © Math As A Second Language All Rights Reserved What happened next is the subject of our next lesson. Hieroglyphics Tally Marks Roman Numerals Place Value Sand Reckoner next pleateau