Real Numbers: Integer Operations

Real Numbers: Adding and Subtracting Integers

• Same signs: Combining two integers with the same sign requires adding the numbers and retaining the original sign.
  • Example: (+3) + (+5) = +8
  • Example: (-2) + (-7) = -9
• Different signs: Combining two integers with differing signs requires subtracting the smaller absolute value from the larger one, with the result adopting the sign of the number possessing the greater absolute value.
  • Example: (+6) + (-4) = +2
  • Example: (-9) + (+3) = -6
• Subtracting a negative number is equivalent to adding a positive number.
  • Example: 5 - (-2) = 5 + 2 = 7

Real Numbers: Multiplying and Dividing Integers

• Same signs: Multiplying or dividing integers with identical signs yields a positive result.
  • Example: (+4) x (+2) = +8
  • Example: (-3) x (-5) = +15
  • Example: (-10) ÷ (-2) = +5
• Different signs: Multiplying or dividing integers with differing signs yields a negative result.
  • Example: (+7) x (-1) = -7
  • Example: (-8) ÷ (+4) = -2
• Order of operations: Follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Real Numbers: Negative Fractions

• Adding/Subtracting: A common denominator is required, then add or subtract the numerators.
  • Example: -1/4 + 3/8 = -2/8 + 3/8 = 1/8
• Multiplying: Multiply numerators and denominators.
  • A negative times a positive is negative, a negative times a negative is positive.
  • Example: (-1/2) x (2/3) = -2/6 = -1/3
• Dividing: Invert the second fraction and multiply. Follow the sign rules for multiplication.
  • Example: (-3/4) ÷ (1/2) = (-3/4) x (2/1) = -6/4 = -3/2

Real Numbers: Negative Decimals

• Adding/Subtracting: Align decimal points and follow integer rules.
• Multiplying: Multiply as usual, then count all decimal places in the original numbers for placement in the answer. Adhere to sign conventions for multiplication.
• Dividing: Adjust the divisor to a whole number by shifting the decimal, mirroring this shift in the dividend. Then divide as usual, following sign conventions.

Real Numbers: Rational and Irrational Numbers

• Rational numbers: Numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Includes whole numbers, integers, terminating decimals, and recurring decimals.
• Irrational numbers: Numbers cannot be written as a simple fraction. Decimals continue without repeating, like π or √2.

Real Numbers: Recurring and Terminating Decimals

• Terminating decimals: Decimals that conclude and can be expressed with a power of 10 as the denominator.
  • Example: 0.25 and 1.5
• Recurring decimals: Decimals with repeating digits and can be written as fractions.
  • Example: 0.333... and 1.272727...

Real Numbers: Index Laws (Multiplication and Division)

• Multiplication: am × an = am+n (e.g., 2³ × 2⁴ = 2⁷)
• Division: am ÷ an = am-n (e.g., 5⁶ ÷ 5² = 5⁴)

Real Numbers: Index Notation (Power of a Power and the Zero Power)

• Power of a Power: (am)n = am×n (e.g., (3²)⁴ = 3⁸)
• Zero Power: a⁰ = 1 (e.g., 7⁰ = 1, (-4)⁰ = 1)

Ratios and Rates: Ratios

• Compares quantities, written as a:b, a to b, or a/b. Ensuring uniform units is imperative for comparison.
  • Example: comparing meters to kilometers requires conversion to same units.

Ratios and Rates: Maps and Scale

• Maps use scale to relate map distance to real distance (e.g., 1 cm represents 10 km). Ratios calculate actual and map distances.

Ratios and Rates: Rates

• Rates compare quantities with different units (e.g., speed (km/hour) and price per item). Divide one quantity by the other to find a rate.

Ratios and Rates: Financial Rates

• Financial rates often use percentages. Simple interest calculation: Interest = Principal x Rate x Time.

Ratios and Rates: Best Buys

• Comparing item costs based on quantity/size identifies the lowest unit price, found via price ÷ quantity.

Time: Elapsed Time

• The duration between start and end, involving addition or subtraction of hours, minutes, or seconds. Requires care when passing AM/PM or day boundaries.

Time: 12 and 24-Hour Time

• 12-hour time: Uses AM (midnight to noon) and PM (noon to midnight).
• 24-hour time: Uses numbers from 00:00 (midnight) to 23:59 (one minute before the next midnight).
  • Converting from 12-hour PM to 24-hour time (after noon), add 12 to the hour (e.g., 3 PM = 15:00).
  • 12 AM is 00:00 in 24-hour time, and 12 PM is 12:00. AM times (except 12 AM) are the same in both systems.

Time: Time Zones

• Different regions operate on different time zones. Account for the differences when traveling or communicating. Earth's rotation dictates that eastern locations experience later times. Add time when moving east and subtract when moving west. The International Date Line adjusts the date when crossed.