1.4 Decimals

Decimals are the backbone of everyday math. From rounding prices to calculating discounts, we use them constantly. This section covers the basics of working with decimals, including arithmetic operations and conversions between different number forms.

Understanding decimals opens doors to more advanced math concepts. We'll explore how decimals fit into the broader world of real numbers, including square roots and irrational numbers. These skills are crucial for tackling more complex algebraic problems down the road.

Decimal Basics

Rounding of decimals

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• Identify place value to round to
  • Tenths, hundredths, thousandths place after decimal point
• Examine digit immediately to right of identified place
  • If digit is 5 or greater, round up by increasing digit in identified place by 1
  • If digit is less than 5, round down by leaving digit in identified place unchanged
• Replace all digits to right of identified place with zeros to complete rounding process

Arithmetic with decimals

• Addition and subtraction
  • Vertically align decimal points in numbers
  • Add or subtract as with whole numbers, ignoring decimal points
  • Place decimal point in result directly below aligned decimal points in original numbers
• Multiplication
  • Multiply numbers as if they were whole numbers, ignoring decimal points
  • Count total number of digits to right of decimal points in factors
  • Place decimal point in product counting from right the same number of places as total count from factors
• Division
  • Shift decimal point in divisor right to make it a whole number
  • Shift decimal point in dividend right same number of places as divisor
  • Divide resulting numbers as whole numbers
  • Place decimal point in quotient directly above decimal point in dividend

Conversions between number forms

• Decimal to fraction
  • Write decimal as numerator over denominator of 1
  • Multiply numerator and denominator by 10 for each place value to right of decimal point
  • Simplify resulting fraction if possible (2.5 = 25/10 = 5/2)
• Fraction to decimal
  • Divide numerator by denominator
  • If division does not terminate, round to desired place value (3/8 = 0.375)
• Decimal to percentage
  • Shift decimal point two places to right and add percent sign (0.25 = 25%)
• Percentage to decimal
  • Remove percent sign and shift decimal point two places to left (40% = 0.40)

Real Numbers and Expressions

Simplification of square roots

• Perfect square has whole number square root
  • Examples: 4, 9, 16, 25, 36
• For perfect square, simplify square root to whole number
  • $\sqrt{25} = 5$
• For non-perfect square, factor out largest perfect square
  • Simplify perfect square factor to whole number
  • Leave non-perfect square factor under square root
  • $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Classification in real numbers

• Natural numbers: positive whole numbers starting from 1 (1, 2, 3, ...)
• Whole numbers: natural numbers and 0 (0, 1, 2, 3, ...)
• Integers: whole numbers and negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers: expressed as fraction $\frac{a}{b}$ with integer $a$ and nonzero integer $b$
  • Includes terminating and repeating decimals (0.5, 0.333...)
• Irrational numbers: cannot be expressed as fraction
  • Non-terminating, non-repeating decimals ($\pi$, $\sqrt{2}$)
• Real numbers encompass all rational and irrational numbers

Number line representations

• For fractions, divide unit interval into equal parts based on denominator
  • Count from 0 to numerator to locate fraction (3/4 is 3 parts out of 4 total)
• For decimals, divide unit interval into 10 parts (tenths)
  • Divide each tenth into 10 parts for hundredths
  • Continue dividing for smaller place values
  • Count from 0 to locate decimal (0.75 is 7 tenths and 5 hundredths)
• Decimal representation on number line reflects the base-10 system

Advanced Decimal Concepts

Scientific notation and significant figures

• Scientific notation expresses numbers as a product of a coefficient and a power of 10
  • Used for very large or very small numbers (e.g., 6.02 × 10^23)
• Significant figures indicate the precision of a measurement
  • All non-zero digits are significant
  • Zeros between non-zero digits are significant
  • Leading zeros are not significant
  • Trailing zeros after a decimal point are significant
• Decimal expansion refers to the representation of a number as a sequence of digits after the decimal point
  • Can be finite (terminating) or infinite (repeating or non-repeating)

Applications

Real-world decimal applications

• Identify given information and unknown quantity in problem
• Determine operation(s) needed to solve (addition, subtraction, multiplication, division)
• Perform decimal arithmetic to calculate answer
• Round answer to appropriate place value based on context (money to cents, measurements to precision)
• Verify solution is reasonable for situation (positive cost, shorter time than original)