Pre-Algebra
Table of Contents
Unit 5 Overview: Decimals
5.1 Decimals
5.2 Decimal Operations
5.3 Decimals and Fractions
5.4 Solve Equations with Decimals
5.5 Averages and Probability
5.6 Ratios and Rate
5.7 Simplify and Use Square Roots

➕pre-algebra review

5.2 Decimal Operations

Citation:

Decimal operations are crucial for everyday math and finance. They involve adding, subtracting, multiplying, and dividing numbers with decimal points. Proper alignment and place value understanding are key to getting accurate results.

Checking decimal calculations is important for accuracy. Estimating results by rounding helps verify if answers make sense. Understanding decimal representation, including scientific notation, is vital for working with very large or small numbers.

Decimal Number Operations

Addition and subtraction of decimals

  • Align the decimal points vertically before performing the operation ensures proper placement of digits
    • Add zeros as placeholders to the right of the decimal point if needed makes the number of decimal places equal (0.5 becomes 0.50)
  • Add or subtract the numbers as if they were whole numbers, ignoring the decimal points simplifies the process
  • Place the decimal point in the answer directly below the decimal points in the problem maintains the correct value

Multiplication with decimal numbers

  • Multiply the numbers as if they were whole numbers, ignoring the decimal points simplifies the calculation
  • Count the total number of digits to the right of the decimal points in all factors determines the placement of the decimal point in the product
  • Place the decimal point in the product so that the number of digits to its right equals the total counted in the previous step ensures the correct magnitude of the result ($2.5 \times 1.2 = 3.00$)

Division involving decimals

  • If the divisor is a decimal, multiply both the divisor and dividend by a power of 10 to make the divisor a whole number simplifies the division process
    • The power of 10 should have as many zeros as there are decimal places in the divisor (0.25 becomes 25 when multiplied by 100)
  • Divide as usual, placing the decimal point in the quotient directly above the decimal point in the dividend maintains the correct placement of the decimal point
  • If the dividend has fewer decimal places than the divisor, add zeros to the right of the last decimal place in the dividend before dividing ensures proper alignment (1.2 ÷ 0.25 becomes 12.0 ÷ 25)

Decimals in money calculations

  • Express money amounts using decimal notation, with the decimal point separating dollars and cents standardizes the representation of currency
    • If a money amount has no cents, add ".00" to the end of the number clarifies the absence of cents ($5 becomes $5.00)
  • Perform the required operation (addition, subtraction, multiplication, or division) following the rules for decimal arithmetic maintains consistency
  • When multiplying or dividing money amounts, round the final answer to the nearest cent if necessary ensures practical results
    • If the result has more than two decimal places, round to the nearest hundredth (second decimal place) follows convention
      • If the third decimal place is 5 or greater, round up; otherwise, round down ($1.255 becomes $1.26, $1.254 becomes $1.25)

Checking Decimal Operation Results

Estimating results for reasonableness

  • Round each decimal number to the nearest whole number or tenth before performing the operation simplifies the calculation
    • For addition and subtraction, round to the same place value for all numbers involved maintains consistency (0.25 + 1.8 becomes 0.3 + 1.8)
    • For multiplication and division, round each number to the nearest whole number provides a rough estimate (2.5 × 1.2 becomes 3 × 1)
  • Perform the operation with the rounded numbers to obtain an estimated result gives an approximate answer
  • Compare the estimated result to the actual calculated result to ensure they are reasonably close verifies the accuracy of the original calculation
  • Follow the order of operations (PEMDAS) when estimating complex expressions with multiple operations ensures consistent results

Decimal Representation and Notation

Understanding decimal numbers

  • Decimal numbers are part of the base-10 number system, which is the foundation of our numerical representation
  • Decimals can be expressed as decimal fractions, showing the relationship between whole numbers and fractional parts
  • The position of each digit in a decimal number determines its value, with each place value being a power of 10

Notation methods

  • Standard notation is the common way of writing decimal numbers in everyday use
  • Scientific notation is useful for expressing very large or very small numbers concisely
    • It represents a number as a product of a decimal between 1 and 10 and a power of 10
    • For example, 0.00345 in scientific notation is 3.45 × 10^-3

Key Terms to Review (29)

Place Value: Place value is a fundamental concept in mathematics that describes the value of a digit based on its position within a number. It is the foundation for understanding and working with whole numbers, decimals, and other numerical representations.
Borrowing: Borrowing is a mathematical operation used to facilitate subtraction when the minuend (the number being subtracted from) does not have a sufficient digit in a particular place value to subtract the corresponding digit in the subtrahend (the number being subtracted). It involves temporarily 'borrowing' from the next higher place value to make the subtraction possible.
Product: The product is the result of multiplying two or more numbers or quantities together. It represents the combined or cumulative effect of the factors involved in the multiplication operation.
Dividend: The dividend is the number that is being divided in a division operation. It represents the total quantity or amount that is to be shared or distributed among a given number of parts or recipients.
Divisor: A divisor is a number that divides another number without leaving a remainder. It is a fundamental concept in mathematics, particularly in the operations of division and factorization, that is essential for understanding various mathematical topics.
Quotient: The quotient is the result of dividing one number by another. It represents the number of times the divisor goes into the dividend, and is the answer to a division problem.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to evaluate an expression. This term is crucial in the context of evaluating, simplifying, and translating expressions, as well as solving equations using various properties of equality.
Decimal Point: The decimal point is a symbol used to separate the whole number part from the fractional part of a number. It is a crucial component in the representation and understanding of decimal numbers, which are essential in various mathematical operations and problem-solving contexts.
Tenths: Tenths refer to the decimal place value that represents one-tenth of a whole. It is a fundamental concept in the understanding of decimals, decimal operations, and solving equations with decimals.
Hundredths: Hundredths refer to the decimal place value that represents a fraction of a whole divided into one hundred equal parts. This term is crucial in understanding decimal notation, decimal operations, and solving equations with decimals.
Thousandths: Thousandths is a decimal place value that represents one part out of one thousand. It is the fourth decimal place, following the ones, tenths, and hundredths places. Thousandths are used to express very small quantities or precise measurements in the context of decimals, decimal operations, and solving equations with decimals.
Rounding Decimals: Rounding decimals is the process of approximating a decimal number to a specified place value, typically to make calculations easier or to present a number in a more concise form. It involves determining the nearest whole number, tenth, hundredth, or thousandth, depending on the desired level of precision.
Terminating Decimal: A terminating decimal is a decimal number that can be expressed as a fraction with a finite number of digits in the denominator. It is a decimal representation that eventually ends or 'terminates' after a finite number of digits.
Repeating Decimal: A repeating decimal is a decimal number in which one or more digits in the decimal part repeat infinitely. This pattern of repeating digits is a characteristic of certain fractions when expressed as a decimal.
Base-10 Number System: The base-10 number system, also known as the decimal number system, is a positional numeral system that uses 10 digits (0-9) to represent all numbers. It is the most widely used number system in the world, and it forms the foundation for decimal notation and arithmetic operations.
Scientific Notation: Scientific notation is a concise way of expressing very large or very small numbers by representing them as a product of a number between 1 and 10 and a power of 10. This format allows for more efficient handling and manipulation of such numbers.
Comparing Decimals: Comparing decimals is the process of determining the relative size or magnitude of decimal numbers. It involves analyzing the digits to the right of the decimal point and using established comparison methods to identify which decimal is greater, less than, or equal to another.
Lining Up Decimal Points: Lining up decimal points refers to the process of aligning the decimal places of numbers in order to perform decimal operations accurately. This ensures that the values are properly positioned and can be added, subtracted, multiplied, or divided correctly.
Multiplication with Decimals: Multiplication with decimals is the process of multiplying two numbers that contain decimal places. It involves applying the standard multiplication algorithm while taking into account the placement of the decimal points in the factors to determine the correct decimal placement in the product.
Division Involving Decimals: Division involving decimals refers to the process of dividing a number that contains a decimal point by another number that also contains a decimal point. This operation is used to find the quotient, or the result of the division, when both the dividend and divisor have decimal places.
Subtraction of Decimals: Subtraction of decimals is the process of subtracting two decimal numbers to find the difference between them. It involves aligning the decimal points and subtracting the digits in the corresponding places, just like subtracting whole numbers.
Money Calculations: Money calculations refer to the mathematical operations and procedures used to manage, manipulate, and analyze financial information, such as income, expenses, budgets, and financial transactions. These calculations are essential for making informed decisions about personal or business finances.
Addition of Decimals: Addition of decimals is the process of combining two or more decimal numbers to find their sum. This operation is a fundamental part of decimal arithmetic and is essential for working with and manipulating decimal quantities.
Cents: Cents are the smallest monetary unit in many countries, representing one-hundredth of a dollar or other currency. They are used in the context of decimal operations to represent fractional values of a whole currency unit.
Estimating Results: Estimating results is the process of approximating or predicting the outcome of a calculation or measurement, often without performing the full, precise computation. It involves using available information and logical reasoning to arrive at a reasonable estimate, rather than a definitive answer.
Decimal Fractions: Decimal fractions are a way of representing fractional quantities using a decimal point. They allow for the expression of values that are not whole numbers, providing a more precise and flexible way to represent quantities compared to common fractions.
Reasonableness: Reasonableness is a concept that refers to the quality of being sensible, rational, and justified based on sound judgment and evidence. It is a fundamental principle applied across various contexts, including legal, ethical, and practical decision-making.
Standard Notation: Standard notation, also known as base-10 or decimal notation, is a way of representing numbers using digits 0-9 and a decimal point to indicate the place value of each digit. It is the most common and widely used system for expressing numerical values.
Factors: Factors are whole numbers that can be multiplied together to produce another number. They play a critical role in various mathematical operations and help in understanding the relationships between numbers. Recognizing factors is essential for simplifying expressions, performing operations, and understanding properties of numbers.