Ratio, Proportion, and Percentage

• Mathematics studies quantity, structure, space, and change
• Topics in math include numbers, formulas, shapes, spaces, quantities, and their changes
• Math problems can be independent from science
• There's no consensus regarding mathematics' exact scope or status

Decimal Numbers

• Decimal numbers use base-10, with symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
• In the decimal system, a digit's value depends on its position
• The digit to the left of the decimal is multiplied by 10⁰ (1), then 10¹ (10), 10², etc.
• The first digit to the right of the decimal is multiplied by 10⁻¹ (0.1), then 10⁻² (0.01), etc.
• Decimal numbers often represent real numbers, including both rational and irrational numbers
• Rational numbers can be fractions of two integers; irrational numbers cannot
• Decimal numbers can be terminating, repeating, or non-repeating/non-terminating
• Terminating and repeating decimals represent rational numbers, whereas non-repeating and non-terminating decimals represent irrational numbers

Rounding Off

• Rounding off approximates a number to a specified number of digits to simplify it
• Whether to round up or down depends on the digit to the right of the last digit you want to keep
• Round up if the digit is 5 or greater, and round down if it's less than 5
• To round to the nearest whole number, consider the tenths place digit
• Focus on the hundredths place digit for rounding to the nearest tenth
• For rounding to the nearest hundredth, look at the thousandths place digit
• Rounding introduces errors but is practical for simplifying results and showing key digits

Scientific Notation

• Scientific notation expresses very large or small numbers conveniently
• A number in scientific notation is the product of a coefficient (1-10) and a power of 10
• Example: 3,000,000 can be written as 3 x 10⁶
• The exponent shows how many places to move the decimal to revert to standard form
• A positive exponent indicates moving the decimal to the right
• A negative exponent indicates moving the decimal to the left
• Scientific notation eases comparing vastly different numbers and performing calculations
• It is widely used in science and engineering
• When multiplying numbers in scientific notation, multiply the coefficients and add the exponents
• When dividing numbers in scientific notation, divide the coefficients and subtract the exponents

Summation Using Sigma

• Summation adds a sequence of numbers, represented concisely using sigma notation (Σ)
• General form: Σᵢ₌ₘⁿ aᵢ (sum of aᵢ from i = m to i = n)
• Σ is the Greek capital letter sigma, representing summation
• i is the index of summation
• m is the lower limit of summation
• n is the upper limit of summation
• aᵢ is the expression being summed
• To evaluate, substitute each value of i from m to n into aᵢ and add the results
• Example: Σᵢ₌₁⁵ i² = 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55
• Summation can represent finite or infinite series sums
• The summation operator adheres to rules and properties like distributive and associative properties
• The summation operator is linear: Σ(caᵢ) = cΣaᵢ and Σ(aᵢ + bᵢ) = Σaᵢ + Σbᵢ where c is a constant
• Summation is used in mathematics, statistics, and physics
• It's used to calculate areas, volumes, probabilities, and other quantities
• Common summation formulas: Σᵢ₌₁ⁿ i = n(n+1)/2, Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6, Σᵢ₌₁ⁿ i³ = [n(n+1)/2]²

Ratio

• A ratio compares two or more quantities
• It indicates how much of one thing there is compared to another
• Ratios can be written in several ways: as a fraction, using a colon, or with the word "to"
• Example: If there are 3 apples and 5 oranges, the ratio of apples to oranges is 3/5, 3:5, or "3 to 5"
• Ratios should be simplified to their lowest terms by dividing all parts of the ratio by their greatest common factor
• Example: The ratio 6:8 can be simplified to 3:4 by dividing both numbers by 2

Proportion

• A proportion is an equation stating that two ratios are equal
• Proportions are used to solve problems involving scaling, similar figures, and direct/inverse relationships
• If a/b = c/d, then the proportion can be cross-multiplied: ad = bc
• This cross-multiplication property is a key tool for solving for unknown values in a proportion
• Example: If 2/x = 6/15, then 2 * 15 = 6 * x, so 30 = 6x, and x = 5
• Direct proportion: as one quantity increases, the other increases proportionally
• Inverse proportion: as one quantity increases, the other decreases proportionally

Percentage

• A percentage is a ratio expressed as a fraction of 100
• The word "percent" means "per hundred"
• Percentages are used to express parts of a whole, changes in quantities, and rates
• To convert a fraction or decimal to a percentage, multiply by 100
• Example: 0.25 = 25%, 1/4 = 25%
• To convert a percentage to a decimal, divide by 100
• Example: 75% = 0.75
• Percentage increase/decrease: calculated as [(New Value - Original Value) / Original Value] * 100
• Example: If a price increases from $20 to $25, the percentage increase is [($25 - $20) / $20] * 100 = 25%
• Calculating percentage of a number: multiply the number by the percentage (as a decimal)
• Example: 20% of 50 is 0.20 * 50 = 10

Dividing Fractions

• Dividing fractions involves multiplying by the reciprocal of the divisor
• The reciprocal of a fraction is obtained by swapping the numerator and the denominator
• Example: The reciprocal of 2/3 is 3/2
• To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction
• (a/b) / (c/d) = (a/b) * (d/c) = (ad) / (bc)
• Example: (1/2) / (3/4) = (1/2) * (4/3) = 4/6 = 2/3
• When dividing mixed numbers, first convert them to improper fractions
• Example: 2 1/2 divided by 1 1/4 becomes 5/2 divided by 5/4, then (5/2) * (4/5) = 20/10 = 2

Radicals

• A radical is a mathematical expression that involves a root, such as a square root, cube root, etc.
• The most common radical is the square root, denoted by √
• The square root of a number 'x' is a value that, when multiplied by itself, equals 'x'
• Example: √9 = 3 because 3 * 3 = 9
• Radicals can be simplified by factoring out perfect squares (or perfect cubes, etc.) from the radicand (the number under the radical)
• Example: √32 = √(16 * 2) = √16 * √2 = 4√2
• Multiplying radicals: √a * √b = √(a*b)
• Example: √2 * √8 = √16 = 4
• Dividing radicals: √a / √b = √(a/b)
• Example: √75 / √3 = √(75/3) = √25 = 5
• Adding and subtracting radicals: can only be done if the radicands are the same
• Combine like terms
• Example: 3√5 + 2√5 = 5√5
• Rationalizing the denominator: Eliminate radicals from the denominator of a fraction
• Multiply both the numerator and denominator by a suitable radical expression that will eliminate the radical in the denominator
• Example: To rationalize 1/√2, multiply both the numerator and denominator by √2: (1/√2) * (√2/√2) = √2 / 2