Understanding Decimal Numbers and Place Value

• Decimal numbers are numbers written in the base-ten number system, using a decimal point to separate the whole number part from the fractional part.

Decimal Place Value

• Each digit in a decimal number has a place value determined by its position relative to the decimal point.
• Moving left from the decimal point, the place values are ones, tens, hundreds, thousands, and so on.
• Moving right from the decimal point, the place values are tenths, hundredths, thousandths, ten-thousandths, and so on.
• Each place value is ten times greater than the place value to its right and one-tenth of the place value to its left.
• For example, in the number 123.45, the digit 1 is in the hundreds place, 2 is in the tens place, 3 is in the ones place, 4 is in the tenths place, and 5 is in the hundredths place.

Reading Decimal Numbers

• The part to the left of the decimal is read as a whole number.
• The decimal point is read as "and".
• The part to the right of the decimal is read as a whole number, followed by the name of the place value of the last digit.
• For example, 123.45 is read as "one hundred twenty-three and forty-five hundredths."
• As another example, 0.678 is read as "six hundred seventy-eight thousandths".

Writing Decimal Numbers

• To write a decimal number from words, write the whole number part.
• Replace "and" with a decimal point.
• Write the decimal part so that the last digit is in the place value indicated by the words.
• Add zeroes as placeholders if necessary.
• For example, "fifty-two and thirty-four hundredths" is written as 52.34.
• As another example, "seven and nine thousandths" is written as 7.009.

Rounding Decimals

• Identify the digit in the place value to which you are rounding (the rounding digit).
• Look at the digit immediately to the right of the rounding digit (the test digit).
• If the test digit is 5 or greater, increase the rounding digit by 1.
• If the test digit is less than 5, do not change the rounding digit.
• Drop all digits to the right of the rounding digit.
• For example, round 3.14159 to the nearest hundredth: the rounding digit is 4, the test digit is 1, so 3.14159 rounded to the nearest hundredth is 3.14.
• As another example, round 12.987 to the nearest tenth: the rounding digit is 9, the test digit is 8, so 12.987 rounded to the nearest tenth is 13.0.

Comparing Decimal Numbers

• Start by comparing the whole number parts of the decimal numbers. If they are different, the number with the larger whole number part is the larger number.
• If the whole number parts are the same, compare the tenths digits. The number with the larger tenths digit is the larger number.
• If the tenths digits are the same, compare the hundredths digits. The number with the larger hundredths digit is the larger number.
• Continue comparing digits in the same place value until you find digits that are different. The number with the larger digit in that place value is the larger number.
• If one number has more digits than the other, you can add zeroes to the end of the shorter number without changing its value (e.g., 0.5 is the same as 0.50).
• For example, compare 4.56 and 4.58: the whole number parts are the same (4), the tenths digits are the same (5), but the hundredths digit of 4.58 (8) is greater than the hundredths digit of 4.56 (6), so 4.58 > 4.56.
• As another example, compare 1.2 and 1.25: the whole number parts are the same (1), the tenths digits are the same (2), so we can add a zero to the end of 1.2 to get 1.20. Comparing 1.20 and 1.25, the hundredths digit of 1.25 (5) is greater than the hundredths digit of 1.20 (0), so 1.25 > 1.2.

Ordering Decimal Numbers

• To order a set of decimal numbers, compare them two at a time, using the rules for comparing decimal numbers.
• Arrange the numbers from smallest to largest or from largest to smallest, as required.
• It can be helpful to write the numbers in a column, aligning the decimal points, and adding zeroes so that all the numbers have the same number of digits after the decimal point.
• For example, order the numbers 3.1, 3.05, and 3.15 from smallest to largest: 3.05 < 3.1 < 3.15.
• As another example, order the numbers 0.6, 0.58, and 0.62 from largest to smallest. Align the decimal points and add a zero to 0.6 to get 0.60. Comparing the numbers, we have 0.62 > 0.60 > 0.58, so the order is 0.62, 0.6, 0.58.