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Lesson 1 Section 2 Place value Positional numeration In our system of writing numbers -- 2,364 for example -- we will see that the powers of 10 are the organizing principle. They are the units. |
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A unit is that by virtue of which
every thing that exists (Euclid, Book VII, Definition 1.) If we count by tens -- 10, 20, 30, 40, 50 -- then 10 is the unit; if we count by hundreds -- 100, 200, 300 -- then 100 is the unit; and if we count by thousands, then 1000 is the unit. Example 1. Let 5 be the unit and count to 30. Answer. "5, 10, 15, 20, 25, 30."
inch.
See Lesson 20: Unit fractions.
And so on. For there is no "42" apart from 42 units, even though we do not say the word units. Positional numeration A system of numeration is a system of naming and writing numbers. The system in worldwide use is based on the powers of 10. To see specifically how, say that we have a very large number of blocks.
Then to name how many, we first group them into as many thousands as we can. Say however that we do not find any thousands, but we do find 3 hundred.
We then group the rest into as many tens as we can. (There will be less than 10 tens, because that would have been another hundred Say that we find: 3 Hundreds + 6 Tens + 4 Ones. This illustrates that, starting with Ones on the right, we have chosen the units to be the powers of 10. There are 3 of that unit (Hundreds), plus 6 of that one (Tens), plus 4 of those (Ones). When we write the number, however, we omit the names of the units and the + signs, and just write "364." "364" means the sum of 3 Hundreds + 6 Tens + 4 Ones. The digit at each place refers to a different unit. We call such a system positional numeration. |
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Example 4. In this number, 139,072,658 the 0 is in which place? Answer. Hundred thousands. For, in each class of three digits, there are Ones, Tens, and Hundreds. 
0 is in the class of thousands and in the Hundreds place. The power of 10 at that position is Hundred thousands. Example 5. In this number, 386,214,035 how many Ten millions are there? (That is, which digit is in the ten millions place?) Answer. 8. For, on counting from the right, the millions are the third group of three digits, 386. The Tens place is the middle one (Ones, Tens, Hundreds). There are 8 Ten millions. Place value versus absolute value In addition to speaking of a digit being "in" a place, we also speak of the place value of the digit itself. In this number, 6,666 each digit has the same absolute or invariable value 6, but a different place value. 6 on the extreme left has the value 6000; the next 6 has the value 600; the next, 60; and the last, 6. Expanded form The symbol for every whole number stands for a sum. 364 = 3 Hundreds + 6 Tens + 4 Ones. (Even the single digits stand for a sum: 5 = 1 + 1 + 1 + 1 + 1.) What is written above is called the expanded form of 364. |
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Example 4. Write 7,328 in expanded form. Answer. Write 7,328 = 7 Thousands + 3 Hundreds + 2 Tens + 8 Ones. In practice, however, it is often more useful to expand the number in this way: 7,328 = 7,000 + 300 + 20 + 8. Units of adjacent place value The following question is to prepare us for the standard written methods of addition and subtraction. The answer follows from the fact that each power of 10 is ten times the number to its right: 1,000 100 10 1 1,000 is made up of ten 100's. 100 is made up of ten 10's. 10 is made up of ten 1's. And so on. |
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Ten 1's can be composed into one 10. Ten 10's can be composed into one 100. Ten 100's can be composed into one 1,000. And so on. Or: 1,000 100 10 1 One 1,000 can be decomposed into ten 100's. One 100 can be decomposed into ten 10's. One 10 can be decomposed into ten 1's. 
1,000 100 10 1 
We will see this when we come to regrouping in addition and subtraction. Rounding off |
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Example 7. Round off 6,528 to the nearest ten. Answer. 6,528 (The wavy equal sign 2 is in the tens place. To round off to the nearest ten, look at the digit to the right: 8 (greater than 5). Therefore, add 1 to the tens place. Replace 8 with 0. Example 8. Round off 6,528 to the nearest hundred. Answer. 6,528 5 is in the hundreds place. To round off to the nearest hundred, look at the digit to the right: 2 (less than 5). Therefore, leave the hundreds place unchanged. Replace 28 with 00. Example 9. Round off 6,528 to the nearest thousand. Answer. 6,528 6 is in the thousands place. To round off to the nearest thousand, look at the digit to the right: 5. Therefore, add 1 to the thousands place. Replace 528 with 000. Example 10. Round off 79,521 to the nearest thousand. Answer. 79,521 9 is in the thousands place. To round off to the nearest thousand, look at the digit to the right: 5. Therefore, add 1 to 79 -- it becomes 80. Replace 521 with 000. To round off decimals, see Lesson 11. At this point, please "turn" the page and do some Problems. or Continue on to the next Section. Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2001-2010 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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) And so we will be left with a number that is less than ten. 
1300
6,5