1. 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 Place Value Sand Reckoner

2. and Beyond... next © Math As A Second Language All Rights Reserved Place Value Place Value Exponential Notation Exponential Notation

3. next © Math As A Second Language All Rights Reserved Even with nouns (denominations) that allow us to keep track of numbers a thousand at a time, with today’s technological development we still run into the same problem as with tally marks. The only difference is that this time, the problem is with the use of 0’s rather than with the use of tally marks. next Exponential Notation

4. As an example that arises in molecular chemistry, the approximate value of Avogadro’s Number, in place value notation, is written as a 6 followed by twenty three 0’s; that is… next © Math As A Second Language All Rights Reserved 600,000,000,000,000,000,000,000. next Even with the commas, is it immediately obvious that there are 23 zeroes in the representation? Or even if it is, is it immediately obvious that the number would be read as 600 sextillion?

5. Thus, as civilization progressed (or, more accurately, as science and technology became more advanced), we found ourselves having to go beyond place value. © Math As A Second Language All Rights Reserved The next step was the birth of exponential notation. In particular we invented a way to abbreviate such products as… 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 next

6. We write the 3 (called the base) and to indicate that we are taking the product of 12 of them, we write 3 12. © Math As A Second Language All Rights Reserved In this notation the 12, which is written in smaller type size above and to the right of the base, is called the exponent. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 next = 3 12 We read 3 12 as “three to the 12 th power” next

7. So what is the point of exponential notation? © Math As A Second Language All Rights Reserved We have now come full circle. We began the evolution from tally marks to place value by showing how the use of tally marks rapidly becomes cumbersome to represent numbers. next

8. © Math As A Second Language All Rights Reserved As an example, while it is difficult at sight to distinguish an array of 29 tally marks from an array of 30 tally marks… | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | …in the language of place value, we can immediately distinguish between much larger numbers, such as 23,456 and 23,457. next

9. In general, it is tedious to convert a number such as 7 23 into an equivalent place-value numeral. We would first have to compute the product of two 7’s, then three 7’s then four 7’s etc, and eventually the product of twenty two 7’s. and then multiply this result by 7. © Math As A Second Language All Rights Reserved However, because of the structure of place value it is easy to compute the powers of 10. next

10. For example, if we wanted to find the place value numeral for 10 4, we would use the following steps… © Math As A Second Language All Rights Reserved 10 1 = 10 next 10 2 = 10 × 10 = 100 10 3 = 10 × 10 × 10 = 1,000 10 4 = 10 × 10 × 10 × 10 = 10,000 next

11. Do you notice the pattern? © Math As A Second Language All Rights Reserved Notes The number of 0’s that follow the 1 is exactly the same as the exponent. 1 next note 1 As a special case, notice if the pattern was to be followed, that 10 0 means a 1 followed by no 0's. In other words 10 0 = 1. 10 4 = 10 × 10 × 10 × 10 = 10,000 next

12. In place value notation, every time we multiply a number by 10, we simply annex a zero at its end. © Math As A Second Language All Rights Reserved Notes For example, 42 × 10 = 420. next In terms of our adjective/noun theme, by annexing a zero to 42 to form 420, the 4 which was modifying 10’s is now modifying 100’s etc.

13. To see this a bit more visually, think in terms of Roman numerals. © Math As A Second Language All Rights Reserved Notes Notice that ten I’s becomes one X, ten X’s become one C, and ten C’s become one M. next Thus, to multiply XXXXII by ten we simply replace each numeral by the next greater one. C C CCXXX X X X I I

14. Returning to Avogadro’s Number, we have a 6 followed by 23 zeroes. Each time we multiply 6 by 10, we annex another 0 to the 6. © Math As A Second Language All Rights Reserved Notes Since there are 23 zeroes after the 6, it means that we multiplied 6 by 10 twenty-three times, that is, by 10 23. next

15. Hence, by the use of exponential notation… © Math As A Second Language All Rights Reserved Notes 600,000,000,000,000,000,000,000 next …can be rewritten in the much more compact form… 6 × 10 23 2 note 2 To grasp how profound the notation 6 X 10 23 is, suppose you wanted to represent this number using tally marks. If you were able to count continuously at rate of 1 billion tally marks per second it would take over 19 million years to complete the task! next

16. Just as it might have been difficult for young children to learn place value, it may be difficult for us to get used to exponential notation. © Math As A Second Language All Rights Reserved Notes next However, once we have internalized the notation, it is then easy to distinguish between two very large numbers.

17. For example, it is difficult to distinguish between a 1 followed by 49 zeroes and a 1 followed by 50 zeroes. © Math As A Second Language All Rights Reserved Notes next However, it is much easier to distinguish between 10 50 and 10 49.

18. The above note is similar to what we saw when place value replaced tally marks. © Math As A Second Language All Rights Reserved Notes next While it was difficult to distinguish between 49 tally marks and 50 tally marks, it was easy to see at a glance that 50 exceeds 49 by 1.

19. While 50 exceeds 49 by 1, 10 50 exceeds 10 49 by a factor of 10. © Math As A Second Language All Rights Reserved Notes next To get from 10 49 to 10 50, we multiply by one more factor of 10. In other words, 10 50 is not just 1 more than 10 49, but rather it is 10 times as great as 10 49.

20. In closing, this saga of the evolution of whole number representation from hieroglyphics to scientific notation, it is naive to view our own generation as the “ultimate in gracious living”. It is certainly natural for us to consider our own generation as the most advanced “state of the art” generation in comparison to past generations, which we might consider technologically more primitive. © Math As A Second Language All Rights Reserved The Beat Goes On…

21. next At the same time, we should not forget that when future generations look back at us, they will find that we were primitive relative to their “state of the art” society. So, as the cliché goes, the beat goes on, and it will be left to future generations to continue the chronicle of human achievement. © Math As A Second Language All Rights Reserved The Beat Goes On…

22. next We will discuss exponential notation in much greater depth in a later presentation. © Math As A Second Language All Rights Reserved However, with respect to the development of our whole number system of enumeration, the present discussion should suffice.

23. next This presentation concludes the Development of Our Number System. © Math As A Second Language All Rights Reserved Hieroglyphics Tally Marks Roman Numerals Place Value Sand Reckoner Exponential Notation