Operations

The Six Operators

The four operations in arithmetic are called addition (+), subtraction (-), multiplication (·), and division (/).Algebra adds two more operations called exponentiation, and radicalization. Together these six operations are called advanced arithmetic. Arithmetic means adding numbers. It is the beginning of all math. Algebra does more than add numbers, it takes numbers and organizes them by using variables, performing functions, graphing, and using formulas.

The six operations have symbols called operators. The operator tells you what operation to do. Every operation has an opposite operation called its reciprocal operation. The word reciprocal and inverse almost mean the same thing, but are used slightly differently depending on context. In general all math from arithmetic to calculus is just adding numbers. Here is a table to show some language and what symbol to use:

Every operation has an inverse operation that takes the answer and works backwards to find the starting number. This can be used to check your work. Checking your work can be used to prove you are correct. Later in geometry you will be asked to show formal proofs.

Addition is adding numbers:

Subtraction is the reciprocal operation of addition so it removes numbers: /[5 - 3 = 2/]
Division is the reciprocal operation of multiplication so it removes groups of the same number: /[15/3 = 5 because 15 – 5 – 5 = 5/]
Rooting is the reciprocal operation of division so it is repeated division: /[²âˆš8 = 2 because 8 ÷ 2 ÷ 2 = 2/]

Sometimes it is important to distinguish what number or symbol you are referring to instead of saying "The number", "squiggle line", or "that thing over there". This is especially important in complex equations that have multiple objects that may refer to the say name. In general math is a binary process where only two numbers are being used at a time. There is the initial value and the number performing the operation. Typically the left number is our "seed" or starting value and the right number is our operating number.

Explanation of Arithmetic notation

Every equation has 5 parts and they are:

For addition subtraction multiplication division exponents, radicals:

Which in English are called :

More formal names that can be used are:

• [augend] + [addend] = [total]
• [minuend] - [subtrahend] = [difference]
• [f₁] · [f₂] = [product]
• [dividend] ÷ [divisor] = [quotient]
• ^[index] √ [radicand] = [root]

In math "notation" means the format of how we write our equations. Notation often means the order and symbols used. This can also be called the "syntax" of math. Different operations use different notation. Different branches of math have different syntax.

• Addition: /[1 + 2 = 3/] 2 is the addend.
• Subtraction: /[10 - 1 = 9/] 1 is the subtrahend.
• Multiplication: /[2 ∙ 3 = 6/] 2 is first factor.
• Division: /[10 ÷ 2 = 5/] 2 is the divisor.
• Exponentiation: /[4² = 16/] 4 is the base and 2 is the exponent (also called power)
• Radicalization: /[√9 = 3/] 9 is the radicand (also called product) and 3 is the root (also called base).

As you can see all the operations have the left most number as the starting point except radicals which start under the radical sign

There are multiple names for the same number being used:

For any operation a value can be called a "number", "value", or a "sum". In addition both augend and addend can be called "addend" or simply "a sum", "sum" or "sums". The result of addition can be called a total or sum. Less common though valid is to call any number a "total", however "sum" is preferred. In any case both "sum" and "total" imply addition or the result of it. The operator for addition uses this symbol " + " and is called "plus sign" or "addition sign".

In subtraction we can say "ten minus one", "one subtracted from ten", "one less than ten", and so on because math allows non fixed use of English phrases due to the highly nuanced nature of the language. This is legal as long as the sentence uttered is logically true. The result of subtraction is called "the difference", "a difference", or even a "total" or "sum". The symbol for subtraction is " - " and is called the subtraction or minus sign.

In arithmetic the raised x symbol " × " is used for multiplication. It is commonly called the "times" symbol. We say "two times three equals six" or "two multiplied by three equals six". In algebra we use the raised dot symbol " ∙ " and language stays the same. Multiplication can also be implied by parentheses or brackets such as 2(3) = 6. Putting variables infront of a number also implies multiplication. When variables are introduced you will see that "2a" = "2 ∙ a".

In division we usually say "ten divided by two equals five". The format has the number to be divided on the left of the division operator followed the number that is doing the dividing. In this case 10 is the dividend (thing which is to be divided), 2 is the divisor (number which is doing the dividing), and 5 is the quotient (number which is the result of the division.) We will learn that there are many symbols for divison such as the obelus " ÷ ", the forward slash " / ", The horizontal fraction bar, the division box, and the pipe symbol " | ". The pipe symbol is rare and mostly used in computing. It reverses the order of dividen and divisor such that 10 ÷ 2 = 5 becomes 2|10 = 5. This is spoken as " ten divided by two equals five" and "two divides ten = equals five". Pipe is not used in Malone Math courses but is in Malone Computer Science courses.

Fractions and ratios also imply division but they are not quite the same and will be discussed deeper later. Fractions call the dividend a "numerator" and the divisor a "denominator". This closely follows standard division and you may call a fraction a "quotient" such that "1/4 = 0.25", where both "1/4" and "0.25" are quotients of "1 ÷ 4".

When a number is being mulitpled by itself over and over we call this a "power" or "exponent" operation. Exponentiation is used to show a number being mulitpled by itself many times. Therefore exponentiation is the way of writing "repeated multiplication of the same number". Example: 2×2×2 = 2³ = 8, 2×2×2×2 = 2⁴ = 16. 2 is the base or root number, the 3 and 4 are called the exponents, and the answers given as 8 and 16 are called the "products". Because the answer to a multiplication problem is called a product we use the same word for the result of an exponent operation. Exponents are also called "powers", which is a bit confusing since the word "power" can be used for the exponent, the answer, and the entire equation! To avoid confusion pick a naming convention and stick with it. Malone Math uses "base", "exponent", and "product" to avoid ambiguity of terms.

Radicals are the result of undoing an exponential equation. Just as division seems harder than multiplcation, radicals are harder than exponents. Since exponents are repeated multiplication, radicals are repeated division. /[2×2×2×2 = 16/] which means: /[2⁴ = 16/] and since: /[16 ÷ 2 ÷ 2 ÷ 2 = 2/] then: /[⁴√16 = 2/]