Decimal numbers are a crucial part of our number system, allowing us to represent parts of a whole with precision. They're used in everyday life for money, measurements, and more. Understanding decimals is key to working with fractions and percentages.

Mastering decimal operations is essential for solving real-world problems. From adding prices to dividing measurements, these skills are used daily. Converting between decimals, fractions, and percentages opens up new ways to interpret and work with numbers.

Understanding Decimal Numbers

Decimal numbers in context

• Decimal numbers represent parts of a whole using place values (decimal notation)
  • Each digit to the right of the decimal point represents a fractional part of a power of 10
    • First digit to the right of the decimal point represents tenths ($\frac{1}{10}$) (0.3 is three tenths)
    • Second digit represents hundredths ($\frac{1}{100}$) (0.07 is seven hundredths)
    • Third digit represents thousandths ($\frac{1}{1000}$) (0.002 is two thousandths), and so on
• Decimals can be written in various forms
  • Standard form (3.14)
  • Word form (three and fourteen hundredths)
  • Expanded form ($3 + \frac{1}{10} + \frac{4}{100}$)
• Interpreting decimals in context
  • Money ($1.50 represents one dollar and fifty cents)
  • Measurements (1.75 meters represents one meter and seventy-five centimeters)
• Decimals are part of the base-10 system, which is fundamental to our number system

Rounding techniques for decimals

• Rounding decimals to a specific place value
  • Identify the place value to which you want to round (tenths, hundredths, etc.)
  • Look at the digit to the right of the identified place value
    • If the digit is 5 or greater, round up by adding 1 to the digit in the identified place value (3.27 rounded to the nearest tenth is 3.3)
    • If the digit is less than 5, keep the digit in the identified place value unchanged (3.42 rounded to the nearest tenth is 3.4)
  • Replace all digits to the right of the identified place value with zeros (3.456 rounded to the nearest hundredth is 3.46)
• Rounding decimals for estimation
  • Round numbers to the nearest whole number, tenth, or hundredth to simplify calculations (3.14 + 2.72 can be estimated as 3 + 3 = 6)
  • Estimate answers to check the reasonableness of calculated results (if the estimate is 6, an answer of 0.586 is likely incorrect)
• The number of significant figures in a measurement determines its precision

Performing Operations with Decimals

Addition and subtraction of decimals

• Align the decimal points vertically (3.14 + 0.7 becomes $\begin{matrix} 3.14 \ +,,0.70 \end{matrix}$)
• Add or subtract digits in each place value column, starting from the rightmost column
• Carry over or borrow as needed, just like in whole number addition and subtraction
• Place the decimal point in the answer directly below the decimal points in the question ($\begin{matrix} 3.14 \ +,,0.70 \ \hline 3.84 \end{matrix}$)

Multiplication and division of decimals

• Multiplication of decimals
  • Multiply the numbers as if they were whole numbers, ignoring the decimal points (3.2 × 1.5 becomes 32 × 15)
  • Count the total number of digits to the right of the decimal points in both factors (3.2 has 1 digit, 1.5 has 1 digit, total is 2 digits)
  • Place the decimal point in the product so that the number of digits to the right of the decimal point equals the total count from the previous step (32 × 15 = 480, with 2 digits to the right of the decimal point, becomes 4.80)
• Division of decimals
  • Move the decimal point in the divisor to the right until it becomes a whole number (1.2 ÷ 0.4 becomes 12 ÷ 4)
  • Move the decimal point in the dividend to the right by the same number of places as in the divisor (1.2 becomes 12)
  • Divide as with whole numbers (12 ÷ 4 = 3)
  • Place the decimal point in the quotient directly above the decimal point in the dividend (1.2 ÷ 0.4 = 3)

Conversion between numerical forms

• Converting decimals to fractions
  • Write the decimal as a fraction over a power of 10 (0.3 = $\frac{3}{10}$, 0.07 = $\frac{7}{100}$)
  • Simplify the fraction if possible ($\frac{3}{10} = \frac{3\div3}{10\div3} = \frac{1}{3}$, $\frac{7}{100} = \frac{7\div7}{100\div7} = \frac{1}{14}$)
• Converting fractions to decimals
  • Divide the numerator by the denominator ($\frac{1}{4} = 1 \div 4 = 0.25$, $\frac{3}{8} = 3 \div 8 = 0.375$)
  • The quotient is the decimal representation of the fraction
• Converting decimals to percentages
  • Move the decimal point two places to the right and add a percent sign (0.75 = 75%, 0.08 = 8%)
• Converting percentages to decimals
  • Remove the percent sign and move the decimal point two places to the left (45% = 0.45, 7% = 0.07)

Advanced decimal concepts

• Scientific notation is used to express very large or very small numbers using decimals and powers of 10
• Decimal operations follow the same order of operations as whole numbers (PEMDAS)
• The precision of a calculation is limited by the least precise measurement used in the calculation