• The commutative property states that the order of multiplication does not matter.
• Mathematically, it can be expressed as: $a \times b = b \times a$
• For example: $3 \times 4 = 4 \times 3 = 12.$

• The associative property states that the grouping of numbers in a multiplication operation does not affect the result. You can change the grouping (i.e., which numbers you multiply together first) without changing the final product.
• Mathematically, it can be expressed as$: (a \times b) \times c = a \times (b \times c)$
• For example, $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24.$

• The distributive property states that multiplying a number by the sum (or difference) of two other numbers is equivalent to multiplying the number by each of those two numbers separately and then adding (or subtracting) the results.
• Mathematically, it can be expressed as: $a \times (b + c) = (a \times b) + (a \times c)$
• For example, $4 \times (2 + 3) = (4 \times 2) + (4 \times 3) = 20.$

• The reciprocal of a number is essentially its multiplicative inverse. It states that division of two numbers can be rewritten as multiplication of one number by the reciprocal of the other number
• $a ÷ b == a × \frac{1}{b}$

• Taking the reciprocal of a number means finding another number such that when you multiply them together, you get 1.
• Example: Taking a reciprocal of 2 is $\frac{1}{2}$.

$\frac{1}{8}$ because $8 \times \frac{1}{8} =1$

• If “a” is a factor of “b,” then “a” divides “b” evenly.

• Express it as $a \times x = b$

  Where:

  • $a$ is the factor.
  • $x$ is some other whole number.
  • $b$ is the original number.

• It means that when you perform the division operation $b ÷ a$, the result is a whole number without any remainder.

• Express a number or algebraic expression as a product of its factors.
• Example: Factorisation of 12 is $2\times2\times3$

• The GCF of two or more numbers is the largest number that evenly divides each of them. It is also known as the greatest common divisor (GCD).
• Example, the GCF of 12 and 18 is 6.

Common factors are factors that two or more numbers share.

A set of numbers that are obtained by multiplying a base number by other whole numbers.

• 15 is a multiple of 3 because 3 can be multiplied by a whole number to give us 15 without any remainder.
• However, 3 is not a multiple of 15 because there is no whole number that can multiply by 15 to get 3 without any remainder.

• Any positive numbers and not inclusive of 0.
• Example(s): 1, 2, 3.

• Number 0 and any positive number and that can be written without fractional component (0,1,2,3)
• Example(s): 0, 1, 2, 3 ..

• Any number (positive and negative and 0.) that can be written without fractional component
• Example(s): -3, -2, -1, 0, 1, 2, 3 ..

• Numbers that can be expressed as fractions. N/D, where N and D are both integers and the D is not 0.
• Example(s): $\frac{1}{2},-\frac{3}{5}$

• Irrational numbers cannot be expressed as fractions with integers in the numerator and denominator, and their decimal representations neither terminate nor repeat.
• Example(s): $\pi$

• Real numbers include all rational and irrational numbers. They represent all possible points on the number line.

• Natural numbers that can be formed by multiplying at least 2 smaller natural numbers. They can be divided evenly by numbers other than 1 and themselves.
• Example(s): 4,6,8,9,10

• Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and themselves.
• Examples(s): 2, 3, 5, 7, 11

2 because that is the only even number that has two exact distinct positive divisors: 1 and itself.

To qualify as a prime number, the number must have exactly two distinct positive divisors: 1 and themselves. Therefore 1 is not a prime number because it is only divisible by itself (1 number) and that is all.

• Ratio is used to compare two quantities / amounts.
• Ratios are used to describe the relative sizes or quantities of different things
• The most common expression is in fraction. In a ratio $a:b$.

It implies a proportional relationship between two quantities. It means for 2 units of $a$ you would have 3 units of $b$.

• Rate is a measure of how one quantity changes in relation to another quantity.
• Rates are often used to describe how something varies or moves with respect to time, distance, or other variables.
• Rates are typically expressed as a ratio of two quantities, where one quantity represents the change or movement.

• Differences:
  • Ratios are used to compare relative size or quantities between two or more things. Whereas rate are used to describe how one quantity changes in relation to another quantity.
  • Ratios are dimensionless and do not have units. Ratios are expressed as a quantity divided by another quantity with units, making them dimensionally consistent)
• Examples:
  • Ratio of 2:3
  • Rates: “Per Unit of Time” / “Miles per Hour”

A percentage can be thought of as a special type of ratio where the denominator is always 100. The numerator of the ratio is part of the whole.

Percentage is a way of expressing a number or ratio as a fraction of 100.

$\frac{10}{20}\times 100$% $= 50$%

• A relationship between two variables where the ratio of their corresponding value is constant.
• Ratio of $2:3$ is the same as the Ratio of $4:6$

• $y=kx$
• Where:
  • y = value of the first variable,
  • x = value of the second variable
  • k = constant of proportionality (What do you do to multiple x to get to y?)

$\frac{3}{5} \times 3 = \frac{9}{5}$

$\frac{1}{3} *\frac{5}{3}= \frac{5}{9}$

Exponent is a repeated operations of multiples

• Power of powers

  $a^m \cdot a^n = a^{m + n}$

• Quotient of powers

  $\frac{a^m}{a^n} = a^{m - n}$

• Zero Exponent

  $a^0 = 1 , \text{(for } a \neq 0 \text{)}$

• Negative Exponent

  $a^{-n} = \frac{1}{a^n}$

• Fractional Exponent

  $a^{\frac{1}{n}} = \sqrt[n]{a}$

• Exponent of 1

  $a^1 = a$

• Exponent of -1

  $a^{-1} = \frac{1}{a}$

• The rule that any nonzero number raised to the power of zero is equal to 1 is a mathematical convention and it is established to maintain consistency
• For example: $a^{m+0} = a^m \cdot a^0 =a^m \cdot1$

When base $x$ raised to the power of a positive exponent, we multiply $x$ by itself $n$ times. Conversely, when we raise to the power of a negative exponent, we inverse the operation by dividing $x$ by itself $n$ times.

$x^{\frac{1}{2}} = \sqrt{x}$

$x^{\frac{a}{b}} = \sqrt[b]{x^a} = (x^a)^\frac{1}{b} = (x^\frac{1}{b})^a$

A square root of a number “x” is a value “y” such that when “y” is multiplied by itself, it equals “x.” In mathematical notation, if $y^2 = x$, then $y$ is the square root of $x$.

Example: The square root of $25$ is $5$ because $5^2 = 25.$

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

This property allows you to split the square root of a product into the product of the square roots of the individual factors.

$\sqrt{12} = \sqrt{(4 * 3)} = \sqrt{4}* \sqrt{3} = 2\sqrt{3}$

$\sqrt{a \div b} = \sqrt{a} \div \sqrt{b}$

This property enables you to split the square root of a quotient into the quotient of the square roots of the numerator and denominator.

$\sqrt{9} \div \sqrt{4} = 3 \div2$