Mathematics Concepts: PEMDAS and Prime Numbers

PEMDAS

• Acronym for the order of operations in mathematics: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
• Operations inside Parentheses are performed first.
• Exponents indicate powers, where ( a^b ) means ( a ) multiplied by itself ( b ) times.
• Multiplication and Division are performed from left to right.
• Addition and Subtraction are also performed from left to right.

Prime Numbers

• Defined as whole numbers greater than 1 that cannot be divided evenly by any number other than themselves and 1.
• The first few prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Lines

• Standard Form: Expressed as ( Ax + By = C ).
• Slope-Intercept Form: Written as ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept.
• Point-Slope Form: Formulated as ( y - y_1 = m(x - x_1) ), used when a point and a slope are known.
• Example equations illustrate how the same line can be represented in different forms.

Difference of Squares

• Formulated as ( a^2 - b^2 = (a + b)(a - b) ).
• Example conversion: ( (x^2 - 16) = 0 ) simplifies to ( (x - 4)(x + 4) = 0 ).

Quadratic Formula

• Used to solve quadratic equations of the form ( ax^2 + bx + c = 0 ).
• The formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
• Example: For the equation ( x^2 + 2x - 3 = 0 ), substituting values provides the solutions ( x = 1 ) and ( x = -3 ).

Divisibility Rules

• 2: A number is divisible if its last digit is 0, 2, 4, 6, or 8.
• 3: A number is divisible if the sum of its digits is divisible by 3 (e.g., 27 → 2 + 7 = 9).
• 4: A two-digit number is divisible if the number formed by its last two digits is divisible by 4 (e.g., 2308 → 08 divided by 4 equals 2).
• 5: A number is divisible if its last digit is 0 or 5.
• 6: A number is divisible if it is divisible by both 2 and 3 (e.g., 18 is divisible by 2 and 3).
• 8: A three-digit number is divisible if the number formed by the last three digits is divisible by 8 (e.g., 2434 → 434 divided by 8 equals 54.25).
• 9: A number is divisible if the sum of its digits is divisible by 9.