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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to read and write decimals. By the end of the module students should understand the meaning of digits occurring to the right of the ones position, be familiar with the meaning of decimal fractions and be able to read and write a decimal fraction.

Section overview

  • Digits to the Right of the Ones Position
  • Decimal Fractions
  • Reading Decimal Fractions
  • Writing Decimal Fractions

Digits to the right of the ones position

We began our study of arithmetic ( [link] ) by noting that our number system is called a positional number system with base ten. We also noted that each position has a particular value. We observed that each position has ten times the value of the position to its right.

10 times 100,000 is the millions position. 10 times 10,000 is the hundred thousands position. 10 times 1,000 is the ten thousands position. 10 times 100 is the thousands position. 10 times 10 is the hundreds position. 10 times 1 is the tens position. 1 is the ones position.

This means that each position has 1 10 the value of the position to its left.

1,000,000 is the millions position. One tenth of 1,000,000 is the hundred thousands. One tenths of 100,000 is the ten thousands. One tenth of 10,000 is the thousands position. One tenth of 1,000 is the thousands. One tenth of 100 is the tens position. One tenth of 10 is the ones position.

Thus, a digit written to the right of the units position must have a value of 1 10 size 12{ { {1} over {"10"} } } {} of 1. Recalling that the word "of" translates to multiplication , we can see that the value of the first position to the right of the units digit is 1 10 size 12{ { {1} over {"10"} } } {} of 1, or

1 10 1 = 1 10 size 12{ { {1} over {"10"} } cdot 1= { {1} over {"10"} } } {}

The value of the second position to the right of the units digit is 1 10 size 12{ { {1} over {"10"} } } {} of 1 10 size 12{ { {1} over {"10"} } } {} , or

1 10 1 10 = 1 10 2 = 1 100 size 12{ { {1} over {"10"} } cdot { {1} over {"10"} } = { {1} over {"10" rSup { size 8{2} } } } = { {1} over {"100"} } } {}

The value of the third position to the right of the units digit is 1 10 size 12{ { {1} over {"10"} } } {} of 1 100 size 12{ { {1} over {"100"} } } {} , or

1 10 1 100 = 1 10 3 = 1 1000 size 12{ { {1} over {"10"} } cdot { {1} over {"10"} } = { {1} over {"10" rSup { size 8{3} } } } = { {1} over {"1000"} } } {}

This pattern continues.

We can now see that if we were to write digits in positions to the right of the units positions, those positions have values that are fractions. Not only do the positions have fractional values, but the fractional values are all powers of 10 10 , 10 2 , 10 3 , size 12{ left ("10","10" rSup { size 8{2} } ,"10" rSup { size 8{3} } , dotslow right )} {} .

Decimal fractions

Decimal point, decimal

If we are to write numbers with digits appearing to the right of the units digit, we must have a way of denoting where the whole number part ends and the fractional part begins. Mathematicians denote the separation point of the units digit and the tenths digit by writing a decimal point . The word decimal comes from the Latin prefix "deci" which means ten, and we use it because we use a base ten number system. Numbers written in this form are called decimal fractions , or more simply, decimals .

millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones are to the left of the decimal point. tenths, hundredths, thousandths, ten thousandths, hundred thousandths, and millionths are to the right of the decimal point.

Notice that decimal numbers have the suffix "th."

Decimal fraction

A decimal fraction is a fraction in which the denominator is a power of 10.

The following numbers are examples of decimals.

  1. 42.6

    The 6 is in the tenths position.

    42 . 6 = 42 6 10 size 12{"42" "." 6="42" { {6} over {"10"} } } {}

  2. 9.8014

    The 8 is in the tenths position.
    The 0 is in the hundredths position.
    The 1 is in the thousandths position.
    The 4 is in the ten thousandths position.

    9 . 8014 = 9 8014 10 , 000 size 12{9 "." "8014"=9 { {"8014"} over {"10","000"} } } {}

  3. 0.93

    The 9 is in the tenths position.
    The 3 is in the hundredths position.

    0 . 93 = 93 100 size 12{0 "." "93"= { {"93"} over {"100"} } } {}

    Quite often a zero is inserted in front of a decimal point (in the units position) of a decimal fraction that has a value less than one. This zero helps keep us from overlooking the decimal point.
  4. 0.7

    The 7 is in the tenths position.

    0 . 7 = 7 10 size 12{0 "." 7= { {7} over {"10"} } } {}

    We can insert zeros to the right of the right-most digit in a decimal fraction without changing the value of the number.
    7 10 = 0 . 7 = 0 . 70 = 70 100 = 7 10 size 12{ { {7} over {"10"} } =0 "." 7=0 "." "70"= { {"70"} over {"100"} } = { {7} over {"10"} } } {}

Reading decimal fractions

Reading a decimal fraction

To read a decimal fraction,
  1. Read the whole number part as usual. (If the whole number is less than 1, omit steps 1 and 2.)
  2. Read the decimal point as the word "and."
  3. Read the number to the right of the decimal point as if it were a whole number.
  4. Say the name of the position of the last digit.

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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