Understanding Decimals: Place Value and Reading

Definition

• Decimals are numbers in the base-10 system, using place value to represent fractions.
• Decimals include a whole number part, a decimal point, and a fractional part.
• The decimal point separates the whole number from the fractional component.

Place Value

• Each digit in a decimal has a specific place value.
• Place values to the left of the decimal point include ones, tens, and hundreds.
• Place values to the right of the decimal point include tenths, hundredths, and thousandths.
• Each position to the right of the decimal point represents a power of 1/10.
• In the number 123.456, 1 is in the hundreds place, 2 is in the tens place, 3 is in the ones place, 4 is in the tenths place, 5 is in the hundredths place, and 6 is in the thousandths place.

Reading Decimals

• The whole number part should be read normally.
• "Point" or "and" should be said at the decimal point.
• Digits after the decimal point are read as a whole number.
• It's important to state the place value of the last digit.
• For example, 3.14 can be read as "three point one four" or "three and fourteen hundredths."
• 0.25 can be read as "zero point two five" or "twenty-five hundredths."

Types of Decimals

• Terminating decimals feature a finite number of digits.
• Repeating decimals have one or more digits repeating infinitely.
• Non-terminating, non-repeating decimals have an infinite number of digits without any repeating pattern, like irrational numbers such as pi.

Converting Fractions to Decimals

• Divide the numerator by the denominator to convert a fraction to a decimal.
• Add zeros after the decimal point in the dividend, if needed.
• Continue dividing until the division terminates, or a repeating pattern is observed.
• For example, 1/4 = 0.25.
• For example, 1/3 = 0.333... which is a repeating decimal.

Converting Decimals to Fractions

• Write the decimal as a fraction with the denominator as a power of 10.
• The power of 10 corresponds to the number of digits after the decimal point.
• Simplify the fraction, if possible.
• For example, 0.75 = 75/100 = 3/4.
• For example, 0.125 = 125/1000 = 1/8.

Comparing Decimals

• First, compare the whole number parts.
• If the whole number parts are equal, compare the digits in the tenths place, the hundredths place, and so on.
• Add trailing zeros, if needed, so that the decimals have the same quantity of decimal places.
• For example, 0.5 > 0.45 because 0.50 > 0.45.
• For example, 1.23 < 1.3 because 1.23 < 1.30.

Rounding Decimals

• Identify the digit in the place value to which the rounding is occurring.
• Look at the digit to the right of the rounding digit.
• Round up the rounding digit if the digit to the right is 5 or more.
• Leave the rounding digit as it is if the digit to the right is less than 5.
• Drop all digits to the right of the rounding digit.
• For example, 3.14159 rounded to two decimal places becomes 3.14.
• For example, 2.765 rounded to the nearest tenth becomes 2.8.

Adding Decimals

• Decimal points should be aligned vertically.
• Add zeros as placeholders, if necessary.
• Add the numbers like whole numbers, starting from the rightmost column.
• Carry over if the sum of a column exceeds 9.
• Place the decimal point in the sum directly below the decimal points in the addends.
• For example, 1.25 + 3.4 + 0.05 = 4.70.

Subtracting Decimals

• Decimal points should be aligned vertically.
• Add zeros as placeholders if needed.
• Subtract as with whole numbers, starting from the rightmost column.
• Borrow from the next column if needed.
• Place the decimal point in the difference directly below the decimal points in the minuend and subtrahend.
• For example, 5.6 - 2.15 = 3.45.

Multiplying Decimals

• Multiply the numbers like whole numbers.
• Count the decimal places in the factors.
• Place the decimal point in the product, ensuring it has the same total of decimal places as the factors.
• For example, 1.5 x 2.5 = 3.75, where 1 + 1 = 2 decimal places.
• For example, 0.2 x 0.3 = 0.06, where 1 + 1 = 2 decimal places.

Dividing Decimals

• Move the decimal point to the right until the divisor is a whole number, if the divisor is a decimal.
• Move the decimal point in the dividend the same number of places to the right.
• Divide as with whole numbers.
• Place the decimal point in the quotient directly above the decimal point in the dividend.
• Add zeros to the dividend as needed to continue the division.
• For example, 4.5 / 1.5 = 45 / 15 = 3.
• For example, 6.25 / 0.5 = 62.5 / 5 = 12.5.

Decimals and Percentages

• Multiply by 100 to convert a decimal to a percentage.
• Divide by 100 to convert a percentage to a decimal.
• For example, 0.75 = 75%.
• For example, 25% = 0.25.

Applications of Decimals

• Decimals are used in measurements of length, weight, height, and temperature.
• Decimals are used in calculations involving money, prices, taxes, and interest rates.
• Science utilizes decimals for scientific notation and calculations involving very small or very large numbers.
• Statistics uses decimals for averages and probabilities.
• Engineering utilizes decimals for precision measurements and design calculations.