Fundamentals and basics

Basic Operations

Both operands in addition are called addends, they result in a sum. The left operand of a subtraction is called the minuend, the right one the subtrahend. The result is the difference. In multiplication both operands are called the factors of a product (the result). Operands and result of a division, from left to right, are called dividend, divisor and quotient.

Addition and multiplication possess commutative and associative properties. The former means the addends or factors can be placed in any order. The latter signifies that they can be grouped in any order. e.g.

Commutative Property

$a + b = b + a$
$a \times b = b \times a$

Associative Property

$(a + b) + c = (a + c) + b$
$(a \times b) \times c = b \times (a \times c)$

Multiplication further possesses the distributive property:

Distributive Property

$a \times (b + c) = (a \times b) + (a \times c)$

Rounding and Estimating

Generally only the place value to the right of the place value we want to round to is relevant in rounding to a value. If it is greater than 5 the place value we are rounding to is increased by 1. e.g

$278,751$ rounded to the ten thousand $\approx 280,000$
$29.584$ rounded to the hundredth $\approx 29.58$

The place values to the right of the one we round to are either omitted or turned to 0s.

When estimating individual values are rounded first before they are added or subtracted.

Remainder

In arithmetic the remainder is what's left after a division operation, e.g $939 / 4 = 234r3$. The modulo operation is used to find the remainder of a division which would otherwise wind up with a decimal. The remainder can be traced from a decimal quotient by the following steps:

$939 / 4 = 234.75$ | subtract the whole number
$234.75 - 234 = 0.75$ | multiply by the original divisor (4)
$r = 0.75 \times 4$
$r = 3$

Average

Given a list of numbers, the average value is found by adding the numbers and dividing the sum by the number of numbers. This is also called the mean.

$n = { 2, 8, 10, 4 }$ average value: $(2 + 8 + 10 + 4) / 4 = 6$

The median is the value in the middle of a sorted set of values. Given a set of ${ 1, 2, 3, 5, 7, 9, 10 }$, the median is the value $5$. Given a set of ${ 0, 1, 5, 6, 8, 14 }$, the median is derived from the set ${ 5, 6 }$ because the numbers of values in the set is even. In the latter case the median is calculated by adding both median values and dividing them by 2: $(5+6)\div2 = 5.5$.

The mode is the most commonly occuring value in a set. A set may have multiple or no modes. The order of values in the set doesn't matter to the mode. Example: The mode of ${ 14, 10, 10, 12, 15, 15, 15, 7, 8 }$ is $15$.

Exponents

$2 \times 2 \times 2 \times 2 \times 2 = 2^5$
The $2$ in $2^5$ is called the base, $5$ is the exponent.

$2^2$ is read as "2 squared"
$2^3$ is read as "2 cubed"
$2^4$ is read as "2 to the power of 4" (all exponents greater than 3 follow this rule)

When paired with parantheses the result of the inside operation has to be multiplied with itself $n-1$ times. e.g $(8 - 6)^2 = 4^2 = 4 \times 4$.

Square Roots

The square root of a number is the inverse of squaring that number. Given $2^2 = 4$, the square root is $\sqrt[2]{4} = 2$. Since a squared number is always positive, the square root of a negative number can't be calculated by ordinary means.

Terms, Expressions and Equalities

Terms are the components of an operations that are added to or subtracted from another. Expressions are made up of two or more terms. Equalities are a set of expressions that are equal in value. Terms may have variables, letters that are meant to be substituted with numbers. Evaluating an expression means to plug in values for its variables and calculating the result. Equalities can be solved for a variable to find a variable's value.

Subscript

The small number to the lower right of a variable is a subscript to distinguish different variables using the same letter.

$x$1 = 5x1​=5, $x$2 = 7 -> $x$1 \ne x2