There are 4 major operations in arithmetic: addition, subtraction, multiplication, and division and there are special names for the numbers used in these operations.
Addition—Consider the equation
a and b are called addends and c is called the sum.
Subtraction—Consider the equation
a is the minuend, b is the subtrahend, and c is the difference.
Multiplication—Consider the equation
a and b are called factors, and c is called the product.
Other ways to write this equation: x
Division—Consider the equation
a is the dividend, b is the divisor, and c is the quotient.
Other ways to write the equation:
Inverse operations—Addition and subtraction are inverse operations because one “undoes” the other. Example: take a number. Now add 3 to it. Next, subtract 3 from that value. You should be at the number you started with. Similarly, multiplication and division are inverse operations of one another.
Inequalities—Sometimes values are not equal, but we still might want to show how they are related. For this, we use inequality symbols. Commonly-used inequality symbols are:
less than
less than or equal to, or at most
greater than
greater than or equal to, or at least
not equal to
Examples of true inequalities:
Note: A combination of numbers, values, and symbols of mathematical operations is called an expression. Once the symbol “=” is involved, the expression becomes an equation. Expressions with inequality symbols are called inequalities.
Examples: is an expression, is an equation, and is an inequality.
Operations with Whole Numbers
Addition and subtraction with whole numbers is as easy as moving right (addition) or left (subtraction) on a number line.
Multiplication of whole numbers is just repeat addition. Example: is just 3 added to itself 4 times:
.
Division is just repeat subtraction. Example: means how many times I can subtract 5 from 15 to get to 0?
, so the answer is 3.
Note: It is generally assumed that you know how to do these operations already, but you should seek extra practice if you’re unsure about any of them.
Operations with Fractions
A fraction () or ratio is just a comparison of two numbers by division. In this example a is the numerator and b is the denominator. If , like in , we say the fraction is improper. You can rewrite as a combination of a whole number and a part: . We call numbers in this form mixed numbers.
It will be important to be able to go back and forth between the two forms. Here is how that’s done:
Improper to Mixed Number:
To write as a mixed number: goes into four times, with remainder , so the answer is .
Mixed Number to Improper:
To write as an improper fraction: times is , add to it and you’ll get . The answer is .
Reciprocal: The reciprocal (or inverse) of a fraction is the fraction .
Addition and Subtraction— Addition and subtraction of fractions rely on one simple concept: common multiples. A multiple of a number can be evenly divided by that number.
So, 5,10, 15, … are all multiples of 5. Note the multiples of 3 are 3, 6, 9, 12, 15, …
Since both 5 and 3 have the multiple 15, we would say 15 is a common multiple of 3 and 5.
Note: There are infinitely many common multiples of 3 and 5, 12 happens to be the least common multiple.
To add (or subtract) two fractions, follow these steps:
Example:
so the new problem is .
3 has multiples 3, 6, 9, 12, 15, …
4 has multiples 4, 8, 12, 16, …
So, the LCD is 12.
Rewrite each with denominator 12: and
Perform the operation:
Note: You will probably need to “reduce this answer to lowest terms” or “simplify” before you are finished. See the section on “Equivalent Forms of Numbers,” above.
Multiplication— Multiplication of fractions is simple. After ensuring no mixed numbers, just multiply the numerators and then multiply the denominators.
Example:
Division— To divide fractions, simply multiply by the reciprocal.
Example:
Lowest Terms— Again, it is important to be able to simplify or reduce your answer to lowest terms. (Reference “Equivalent Forms of Numbers” above for a quick refresher on reducing fractions.) Usually, answers are expected to be in lowest terms, but be aware that an unreduced answer might still be correct if the reduced choice is not present.
Operations with Decimals
A decimal number is just another way to represent a fraction or mixed number using the base of our number system, 10. We’ve already gone over how to convert between the two.
Addition and Subtraction— To add (or subtract) decimals, simply line up the decimal points and add vertically. Note: Lining up the decimal points in addition and subtraction is critical.
Example:
Multiplication— To multiply decimals, multiply vertically like normal. Then count up the total number of digits to the right of the decimal point in the two factors. Count from the right that many digits in the product and place your decimal point there.
Example:
(Note: we moved two places from the right and put the decimal point between 3 and 2.)
Division— To divide decimals follow these steps:
Example:
Operations with Positive and Negative Numbers
A negative number is a real number less than 0. Negative numbers lie to the left of 0 on the number line. Reference this image:
Addition and Subtraction— In general, addition means you’re moving to the right on the number line, whereas subtraction means you’re moving left. For the example 3 - 5, start at 3 on the number line and move 5 to the left. You’ll end up at -2.
Adding a Negative is just subtraction. Example:
Subtracting a Negative is just addition. Example:
Note: Addition is commutative (remember: a + b = b + a). In the last example - 3 + 8, instead of thinking about starting at -3 on the number line and moving 8 to the right, turn it into .
Multiplication and Division— For multiplication and division follow this rule: If the signs are the same, the answer is positive. Otherwise the answer is negative.
Example: because the signs are the same (both negative).
Example: because the signs aren’t the same (positive and negative).
If I give you the problem , I should assume that you and I would simplify the expression to the same value. Let’s examine two ways to do this problem:
You:
Me:
You got 9 and I got 7. So, who’s right? We’ve developed a system called the Order of Operations to solve this debate. I was wrong because I didn’t follow the order of operations.
Remember to follow the order of operations (PEMDAS):
Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.
Good way to remember: Please Excuse My Dear Aunt Sally