Decimal Numbers: Rounding & Operations

Definition: Adjusting a decimal number to a nearby value for simplicity.
Rules:
- Identify the place value to round to (tenths, hundredths, etc.).
- Look at the digit immediately to the right:
  - If it's 5 or more, round up.
  - If it's less than 5, round down.
Examples:
- 3.276 rounded to the nearest hundredth → 3.28
- 4.582 rounded to the nearest tenth → 4.6

Addition:
- Align decimal points before adding.
- Fill in with zeros if necessary.
Subtraction:
- Align decimal points as with addition.
- Borrowing may be necessary when digits are smaller.
Multiplication:
- Multiply as if there are no decimals.
- Count total decimal places in the factors and place the decimal in the product accordingly.
Division:
- Move the decimal point in the divisor to make it a whole number.
- Move the decimal point in the dividend the same number of places.
- Proceed with long division.

Definition: A way to express numbers using a decimal point to separate the whole number from the fractional part.
Structure:
- Whole number part (left of the decimal).
- Fractional part (right of the decimal).
Examples:
- 7.25 (7 is the whole number, 25 is the fractional part).
Place Values:
- Tenths (0.1), Hundredths (0.01), Thousandths (0.001), etc.
Significance:
- Allows for precise representation of values, especially in measurements and financial calculations.

Rounding adjusts a decimal to a nearby value for ease of use.
Key rules for rounding:
- Choose the place value to round to (e.g., tenths, hundredths).
- Check the digit immediately right of the target place:
  - A digit of 5 or more increases the target digit by one.
  - A digit less than 5 retains the target digit.
Example of rounding:
- 3.276 rounded to the nearest hundredth is 3.28.
- 4.582 rounded to the nearest tenth is 4.6.

Addition:
- Align decimal points before performing addition.
- Pad with zeros where necessary for proper alignment.
Subtraction:
- Similar alignment method as addition.
- Borrowing may be needed when subtracting smaller digits.
Multiplication:
- Multiply numbers without considering decimals initially.
- Count total decimal places in both numbers to determine the decimal placement in the result.
Division:
- Adjust the divisor to become a whole number by moving its decimal.
- Shift the dividend's decimal by the same number of places.
- Execute long division as normal thereafter.

Decimal notation represents numbers using a decimal point, separating whole and fractional components.
Structure consists of:
- Whole number part (left of the decimal point).
- Fractional part (right of the decimal point).
Example of decimal notation:
- In 7.25, 7 is the whole number and 25 is the fractional part.
Place Values include:
- Tenths (0.1), Hundredths (0.01), Thousandths (0.001), among others.
Significance lies in providing precise value representation, crucial in measurements and financial calculations.

Decimal numbers utilize a base-10 numeral system, incorporating whole numbers and fractions divided by a decimal point.

Decimal numbers are formatted as a.bc:
- a represents the whole number portion.
- . serves as the decimal point.
- bc denotes the fractional component.
In the example 12.34, "12" is the whole part, while "34" is the fractional part.

Each digit in a decimal number has distinct place values:
- Whole numbers are categorized into units, tens, hundreds, etc.
- Fractions are categorized into tenths, hundredths, thousandths, etc.
Example with 12.345:
- 1 corresponds to 10 (tens).
- 2 corresponds to 2 (units).
- 3 corresponds to 0.3 (tenths).
- 4 corresponds to 0.04 (hundredths).
- 5 corresponds to 0.005 (thousandths).

Addition/Subtraction:
- Align the decimal points.
- Execute calculations as with whole numbers.
Multiplication:
- Multiply the numbers as if they were whole.
- Count all decimal places from both factors and adjust the product's decimal point accordingly.
Division:
- Shift the decimal point in the divisor to create a whole number.
- Shift the decimal point in the dividend the same number of places.
- Proceed with division as with whole numbers.

Rounding adjusts decimal numbers to a specific decimal position:
- If the subsequent digit is 5 or greater, the number rounds up.
- If it is less than 5, the number rounds down.

Fractions to Decimals: Achieved by dividing the numerator by the denominator.
Decimals to Fractions:
- Convert the decimal to a fraction (e.g., 0.75 equals 75/100).
- Simplification is necessary when possible.

Widely applied in finance (e.g., currency), measurements, statistics, and various scientific contexts.
Fundamental for accurate calculations in everyday scenarios.