Place Value of Whole Numbers and Decimals
Whole Numbers and NumerationThe Hindu-Arabic system of enumeration is described as a base-ten positional number system or a decimal system. It begins with special symbols called digits to represent the first nine counting numbers—1, 2, 3, 4, 5, 6, 7, 8, 9. There is also a very important tenth digit, 0, that is used to represent an empty column. Hindus are credited with the development of this system, and Arabs with introducing it to Western Europe.
The importance and power of a positional numeration system cannot be understated. It is truly one of the most significant inventions of civilization. All standard computational algorithms for whole numbers and decimals that students learn in elementary school are based on and work because of the base-ten positional numeration system.
For whole numbers, the digit farthest to the right is in the ones place. Moving to the left, each digit has a place value equal to 10 times that of the digit to its right. This can be shown in a place-value chart. The number 347,052 can be understood by reading the following place-value chart.
Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
---|---|---|---|---|---|
10 × 10,000 or 10^{5} 100,000 |
10 × 1,000 or 10^{4} 10,000 |
10 × 100 or 10^{3} 1,000 |
10 × 10 or 10^{2} 100 |
10 × 1 or 10^{1} 10 |
1 10^{0} 1 |
3 | 4 | 7 | 0 | 5 | 2 |
The digit 7 is in the thousands place. Its value is 7 thousands. The digit 5 is in the tens place. Its value is 5 tens. When writing numbers in word form, hyphens are used for numbers between 21 and 99 (except those ending in 0).
Numerals are written symbols for numbers. In numerals greater than 999, the digits are grouped into periods of three that are set off by commas. The ones period consists of the digits in the hundreds, tens, and ones places. The thousands period consists of the three places to the left of the ones period. The millions period consists of the three places to the left of the thousands period. The numerals in each period are read as a three-digit number, and the period name is then added (except for the ones period). The place-value chart can be extended to include greater numbers.
Billions | Millions | Thousands | Ones | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Hundreds | Tens | Ones | Hundreds | Tens | Ones | Hundreds | Tens | Ones | Hundreds | Tens | Ones |
10^{11} | 10^{10} | 10^{9} | 10^{8} | 10^{7} | 10^{6} | 10^{5} | 10^{4} | 10^{3} | 10^{2} | 10^{1} | 10^{0} |
9 | 8 | 6 | 5 | 2 | 6 | 0 | 4 | 1 | 3 | 7 | 5 |
Thus, for example, 986,526,041,375 is read as nine hundred eighty-six billion, five hundred twenty-six million, forty-one thousand, three hundred seventy-five (with no mention of the ones period).
Comparing and Rounding Whole Numbers
An important application of place value is its use in comparing numbers. To compare 51,432 and 9,567, students note that the first number has five digits and the second number has four digits and conclude that 51,432 > 9,567. This method is justified by the fact that for whole numbers the least 2-digit number (10) is greater than the greatest 1-digit number (9); the least 3-digit number (100) is greater than the greatest 2-digit number (99); and so on.
To compare two whole numbers with the same number of digits, students begin by comparing digits in each place, starting at the left. The number with the greater digit in that place is the greater number. For example, when comparing 21,487 and 21,612, the digits in the ten thousands and thousands places are the same, but the digits in the hundreds places are different. The fact that 6 > 4 implies that 21,612 > 21,487. Students should also note that 6 > 4 can be written as 4 < 6, which implies that 21,487 < 21,612.
The concept of place value is also applied when rounding numbers. When rounding a number N to the nearest hundred, students determine the number closest to N that has all zeros to the right of the hundreds place. For example, 5,378 rounded to the nearest hundred will be either 5,300 or 5,400. Since 5,400 = 5,378 + 22 and 5,378 = 5,300 + 78, students will note that 5,378 is closer to 5,400 than to 5,300. When both differences are equal, the rounding rule calls for choosing the greater value. Thus 5,378 rounded to the nearest hundred is 5,400.
When rounding a whole number, all places to the right of the rounded place are changed to zeros. For example, 238,574 rounded to the nearest thousand is 239,000; to the nearest hundred, 238,600; and to the nearest ten, 23,570.
The number line makes it easy to visualize why 5,378 rounds to 5,400: the point representing 5,378 is closer to the tick mark at 5,400 than to the tick mark at 5,300.
Teaching Model 1.1: Place Value Through Hundred Thousands