Numbers and Operations
Introduction to Arithmetic Operations
Basic OperationsThe basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
Learning ObjectivesCalculate the sum, difference, product, and quotient of positive whole numbers
- The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
- The basic arithmetic properties are the commutative, associative, and distributive properties.
- associative: Referring to a mathematical operation that yields the same result regardless of the grouping of the elements.
- commutative: Referring to a binary operation in which changing the order of the operands does not change the result (e.g., addition and multiplication).
- product: The result of multiplying two quantities.
- quotient: The result of dividing one quantity by another.
- sum: The result of adding two quantities.
- difference: The result of subtracting one quantity from another.
The Four Arithmetic Operations
AdditionAddition is the most basic operation of arithmetic. In its simplest form, addition combines two quantities into a single quantity, or sum. For example, say you have a group of 2 boxes and another group of 3 boxes. If you combine both groups together, you now have one group of 5 boxes. To represent this idea in mathematical terms:
SubtractionSubtraction is the opposite of addition. Instead of adding quantities together, we are removing one quantity from another to find the difference between the two. Continuing the previous example, say you start with a group of 5 boxes. If you then remove 3 boxes from that group, you are left with 2 boxes. In mathematical terms:
MultiplicationMultiplication also combines multiple quantities into a single quantity, called the product. In fact, multiplication can be thought of as a consolidation of many additions. Specifically, the product of
is the result of
times. For example, one way of counting four groups of two boxes is to add the groups together:
However, another way to count the boxes is to multiply the quantities:
Note that both methods give you the same result—8—but in many cases, particularly when you have large quantities or many groups, multiplying can be much faster.
DivisionDivision is the inverse of multiplication. Rather than multiplying quantities together to result in a larger value, you are splitting a quantity into a smaller value, called the quotient. Again, to return to the box example, splitting up a group of 8 boxes into 4 equal groups results in 4 groups of 2 boxes:
The Basic Arithmetic Properties
Commutative PropertyThe commutative property describes equations in which the order of the numbers involved does not affect the result. Addition and multiplication are commutative operations:
Subtraction and division, however, are not commutative.
Associative PropertyThe associative property describes equations in which the grouping of the numbers involved does not affect the result. As with the commutative property, addition and multiplication are associative operations:
Once again, subtraction and division are not associative.
Distributive PropertyThe distributive property can be used when the sum of two quantities is then multiplied by a third quantity.
Negative NumbersArithmetic operations can be performed on negative numbers according to specific rules.
Learning ObjectivesCalculate the sum, difference, product, and quotient of negative whole numbers
- The addition of two negative numbers results in a negative; the addition of a positive and negative number produces a number that has the same sign as the number of larger magnitude.
- Subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude, while subtracting a negative number yields the same result as adding a positive number.
- The product of one positive number and one negative number is negative, and the product of two negative numbers is positive.
- The quotient of one positive number and one negative number is negative, and the quotient of two negative numbers is positive.
The Four Operations
AdditionThe addition of two negative numbers is very similar to the addition of two positive numbers. For example:
The underlying principle is that two debts—negative numbers— can be combined into a single debt of greater magnitude.
When adding together a mixture of positive and negative numbers, another way to write the negative numbers is as positive quantities being subtracted. For example:
Here, a credit of 8 is combined with a debt of 3, which yields a total credit of 5. However, if the negative number has greater magnitude, then the result is negative:
Here, a debt of 2 is combined with a credit of 7. The credit has greater magnitude than the debt, so the result is positive. But if the credit is less than the debt, the result is negative:
SubtractionSubtracting positive numbers from each other can yield a negative answer. For example, subtracting 8 from 5:
Subtracting a positive number is generally the same as adding the negative of that number. That is to say:
Similarly, subtracting a negative number yields the same result as adding the positive of that number. The idea here is that losing a debt is the same thing as gaining a credit. Therefore:
MultiplicationWhen multiplying positive and negative numbers, the sign of the product is determined by the following rules:
- The product of two positive numbers is positive.The product of one positive number and one negative number is negative.
- The product of two negative numbers is positive.
This is simply because adding −2 together three times yields −6:
The idea again here is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:
DivisionThe sign rules for division are the same as for multiplication.
- Dividing two positive numbers yields a positive number.
- Dividing one positive number and one negative number yields a negative number.
- Dividing two negative numbers yields a positive number.
If the dividend and the divisor have the same sign, that is to say, the result is always positive. For example:
Additional ConsiderationsThe basic properties of addition (commutative, associative, and distributive) also apply to negative numbers. For example, the following equation demonstrates the distributive property:
FractionsA fraction represents a part of a whole and consists of an integer numerator and a non-zero integer denominator.
Learning ObjectivesCalculate the result of operations on fractions
- Addition and subtraction of fractions require "like quantities"—a common denominator. To add or subtract fractions containing unlike quantities (e.g. adding quarters to thirds), it is necessary to convert all amounts to like quantities.
- Multiplication of fractions requires multiplying the numerators by each other and then the denominators by each other. A shortcut is to use the cancellation strategy, which reduces the numbers to the smallest possible values prior to multiplication.
- Division of fractions involves multiplying the first number by the reciprocal of the second number.
- numerator: The number that sits above the fraction bar and represents the part of the whole number.
- reciprocal: A fraction that is turned upside down so that the numerator and denominator have switched places.
- denominator: The number that sits below the fraction bar and represents the whole number.
- fraction: A ratio of two numbers—a numerator and a denominator—usually written one above the other and separated by a horizontal bar.
, consists of an integer numerator (the top number) and a non-zero integer denominator (the bottom number). The numerator represents a certain number of equal parts of the whole, and the denominator indicates how many of those parts are needed to make up one whole. An example can be seen in the following figure, in which a cake is divided into quarters:
Adding Like QuantitiesThe first rule of adding fractions is to start by adding fractions that contain like denominators—for example, multiple fourths, or quarters. A quarter is represented by the fraction
, where the numerator, 1, represents the single quarter and the denominator, 4, represents the number of quarters it takes to make a whole, or one dollar.
Imagine one pocket containing two quarters, and another pocket containing three quarters. In total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:
Adding Unlike QuantitiesTo add fractions that contain unlike denominators (e.g. quarters and thirds), it is necessary to first convert all amounts to like quantities, which means all the fractions must have a common denominator. One easy way to to find a denominator that will give you like quantities is simply to multiply together the two denominators of the fractions. (It is important to remember that each numerator must also be multiplied by the same value its denominator is being multiplied by in order for the fraction to represent the same ratio.)
For example, to add quarters to thirds, both types of fractions are converted to twelfths:
This method can be expressed algebraically as follows:
This method always works. However, sometimes there is a faster way—a smaller denominator, or a least common denominator—that can be used. For example, to add
, the denominator 48 (the product of 4 and 12, the two denominators) can be used—but the smaller denominator 12 (the least common multiple of 4 and 12) may also be used.
Adding Fractions to Whole NumbersWhat if a fraction is being added to a whole number? Simply start by writing the whole number as a fraction (recall that a whole number has a denominator of
), and then continue with the above process for adding fractions.
SubtractionThe process for subtracting fractions is, in essence, the same as that for adding them. Find a common denominator, and change each fraction to an equivalent fraction using that common denominator. Then, subtract the numerators. For instance:
To subtract a fraction from a whole number or to subtract a whole number from a fraction, rewrite the whole number as a fraction and then follow the above process for subtracting fractions.
MultiplicationUnlike with addition and subtraction, with multiplication the denominators are not required to be the same. To multiply fractions, simply multiply the numerators by each other and the denominators by each other. For example:
If any numerator and denominator shares a common factor, the fractions can be reduced to lowest terms before or after multiplying. For example, the resulting fraction from above can be reduced to
because the numerator and denominator share a factor of 6. Alternatively, the fractions in the initial equation could have been reduced, as shown below, because 2 and 4 share a common factor of 2 and 3 and 3 share a common factor of 3:
To multiply a fraction by a whole number, simply multiply that number by the numerator of the fraction:
A common situation where multiplying fractions comes in handy is during cooking. What if someone wanted to "half" a cookie recipe that called for
of a cup of chocolate chips? To find the proper amount of chocolate chips to use, multiply
. The result is
, so the proper amount of chocolate chips is
of a cup.
DivisionThe process for dividing a number by a fraction entails multiplying the number by the fraction's reciprocal. The reciprocal is simply the fraction turned upside down such that the numerator and denominator switch places. For example:
To divide a fraction by a whole number, either divide the fraction's numerator by the whole number (if it divides simply):
or multiply the fraction's denominator by the whole number:
Complex FractionsA complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.
Learning ObjectivesSimplify complex fractions
- Complex fractions include numbers such as and, where the numerator, denominator, or both include fractions.
- Before solving complex rational expressions, it is helpful to simplify them as much as possible.
- The "combine-divide method" for simplifying complex fractions entails (1) combining the terms in the numerator, (2) combining the terms in the denominator, and finally (3) dividing the numerator by the denominator.
- complex fraction: A ratio in which the numerator, denominator, or both are themselves fractions.
are complex fractions. When dealing with equations that involve complex fractions, it is useful to simplify the complex fraction before solving the equation.
The process of simplifying complex fractions, known as the "combine-divide method," is as follows:
- Combine the terms in the numerator.
- Combine the terms in the denominator.
- Divide the numerator by the denominator.
Example 1Let's apply this method to the first complex fraction presented above:
Since there are no terms that can be combined or simplified in either the numerator or denominator, we'll skip to Step 3, dividing the numerator by the denominator:
From previous sections, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Therefore, we use the cancellation method to simplify the numbers as much as possible, and then we multiply by the simplified reciprocal of the divisor, or denominator, fraction:
Therefore, the complex fraction
Example 2Let's try another example:
Start with Step 1 of the combine-divide method above: combine the terms in the numerator. You'll find that the common denominator of the two fractions in the numerator is 6, and then you can add those two terms together to get a single fraction term in the larger fraction's numerator:
Let's move on to Step 2: combine the terms in the denominator. To do so, we multiply the fractions in the denominator together and simplify the result by reducing it to lowest terms:
Let's turn to Step 3: divide the numerator by the denominator. Recall, again, that dividing by a fraction is the same as multiplying by the reciprocal of that fraction:
Finally, simplify the resultant fraction:
Introduction to ExponentsExponential form, written
, represents multiplying the base
- Describe exponents as representing repeated multiplication
- Exponentiation is a mathematical operation that represents repeated multiplication. The exponent in the expressionrepresents the number of times the baseis multiplied by itself. For example, the expressionrepresents.
- Any number raised to the exponent is the number itself. For example,.
- Any nonzero number raised to the exponent 0 is 1. That is to say, .
- base: A number raised to the power of an exponent.
- exponent: The power to which a number, symbol, or expression is to be raised. For example, the 3 in .
in the expression
represents the number of times the base
is multiplied by itself.
For example, the expression
. Here, the exponent is 3, and the expression can be read in any of the following ways:
- raised to the 3rd power
- raised to the power of
- raised by the exponent of
Some exponents have their own unique pronunciations. For example,
is usually read as "
Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
Positive Integer ExponentsNow that we understand the basic idea, let's practice simplifying some exponential expressions.
Example 1Let's look at an exponential expression with 2 as the base and 3 as the exponent:
This means that the base 2 gets multiplied by itself 3 times:
Example 2Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent:
This means that the base 3 gets multiplied by itself 5 times:
Exponents of 0 and 1Any number raised by the exponent
is the number itself. That is to say,
Any nonzero number raised by the exponent 0 is 1. That is to say,
. For example,
The Order of OperationsThe order of operations is an approach to evaluating expressions that involve multiple arithmetic operations.
Learning ObjectivesDifferentiate between correct and incorrect uses of the order of operations
- The order of operations prevents ambiguity in mathematical expressions.
- The order of operations is as follows: 1) simplify terms inside parentheses or brackets, 2) simplify exponents and roots, 3) perform multiplication and division, 4) perform addition and subtraction.
- Multiplication and division are given equal priority, as are addition and subtraction. This means that multiplication and division operations (and similarly addition and subtraction operations) can be performed in the order in which they appear in the expression.
- A helpful mnemonic to remember the order of operations is PEMDAS, sometimes expanded to "Please Excuse My Dear Aunt Sally."
- mathematical operation: An action or procedure that produces a new value from one or more input values.
For example, when faced with the expression
, how do you proceed?
One option is:
The other option is:
Which one is the correct order of operations?
In order to be able to communicate using mathematical expressions, we must have an agreed-upon order of operations so that each expression is unambiguous. For the above expression, for example, all mathematicians would agree that the correct answer is 10.
The order of operations used throughout mathematics, science, technology, and many computer programming languages is as follows:
- Simplify terms inside parentheses or brackets
- Simplify exponents and roots
- Perform multiplication and division
- Perform addition and subtraction
These rules means that within a mathematical expression, the operation ranking highest on the list should be performed first. Multiplication and division are of equal precedence (tier 3), as are addition and subtraction (tier 4). This means that multiplication and division operations (and similarly addition and subtraction operations) can be performed in the order in which they appear in the expression.
Let's evaluate a few expressions using the order of operations.
In this expression, the following operations are taking place: exponentiation, subtraction, multiplication, and addition. Following the order of operations, we simplify the exponent first and then perform the multiplication; next, we perform the subtraction, and then the addition:
Here we have an expression that involves subtraction, parentheses, multiplication, addition, and exponentiation. Following the order of operations, we simplify the expression within the parentheses first and then simplify the exponent; next, we perform the subtraction and addition operations in the order in which they appear in the expression:
A Note on Equal PrecedenceSince multiplication and division are of equal precedence, it may be helpful to think of dividing by a number as multiplying by the reciprocal of that number. Thus
. In other words, the quotient of 3 and 4 equals the product of 3 and
Similarly, as addition and subtraction are of equal precedence, we can think of subtracting a number as the same as adding the negative of that number. Thus
. In other words, the difference of 3 and 4 equals the sum of positive three and negative four.
With this understanding, think of
as the sum of 1, negative 3, and 7, and then add these terms together. Now that you've reframed the operations, any order will work:
The important thing is to keep the negative sign with any negative number (here, the 3).
MnemonicsIn the United States, the acronym PEMDAS is a common mnemonic for remembering the order of operations. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally."
This mnemonic can be misleading, however, because the "MD" implies that multiplication must be performed before division and the "AS" that addition must be performed before subtraction, rather than acknowledging their equal precedence. To illustrate why this is a problem, consider the following:
This expression correctly simplifies to 9. However, if you were to add together 2 and 3 first, to give 5, and then performed the subtraction, you would get 5 as your final answer, which is incorrect. To avoid this mistake, is best to think of this problem as the sum of positive ten, negative three, and positive two.
To avoid this confusion altogether, an alternative way to write the mnemonic is:
Or, simply as PEMA, where it is taught that multiplication and division inherently share the same precedence and that addition and subtraction inherently share the same precedence. This mnemonic makes the equivalence of multiplication and division and of addition and subtraction clear.