Applied Math: Operations

Operations
An operation is simply a mathematical process that takes some value(s) as input(s) and produces an output. Elementary operations are often written in the following form: value operation value.
For instance, in the expression  the values are 1 and 2 and the operation is addition. Performing the operation gives the output of 3. In this way we can say that  and 3 are equal, or .

Addition
Addition increases the value of one quantity by the value of another quantity (both called addends). For example, .
The result is called the sum. With addition, the order does not matter, .
When adding signed numbers, if the signs are the same simply add the absolute values of the addends and apply the original sign to the sum.
For example,  and .
When the original signs are different, take the absolute values of the addends and subtract the smaller value from the larger value, then apply the original sign of the larger value to the difference. For instance,  and
.

Subtraction
Subtraction is the opposite operation to addition; it decreases the value of one quantity (the minuend) by the value of another quantity (the subtrahend). For example, .
The result is called the difference. Note that with subtraction, the order does matter,.
For subtracting signed numbers, change the sign of the subtrahend and then follow the same rules used for addition. For example, .

Multiplication
Multiplication can be thought of as repeated addition. One number (the multiplier) indicates how many times to add the other number (the multiplicand) to itself.

For example, .

With multiplication, the order does not matter:  or , either way the result (the product) is the same.
If the signs are the same the product is positive when multiplying signed numbers.

For example,  and .
If the signs are opposite, the product is negative. For example,  and
.

When more than two factors are multiplied together, the sign of the product is determined by how many negative factors are present. If there are an odd number of negative factors then the product is negative, whereas an even number of negative factors indicates a positive product.

For instance,  and .

Division
Division is the opposite operation to multiplication; one number (the divisor) tells us how many parts to divide the other number (the dividend) into. The result of division is called the quotient.
For example, ; if 20 is split into 4 equal parts, each part is 5. With division, the order of the numbers does matter, .
The rules for dividing signed numbers are similar to multiplying signed numbers. If the dividend and divisor have the same sign, the quotient is positive. If the dividend and divisor have opposite signs, the quotient is negative.

For example, .

Parentheses
Parentheses are used to designate which operations should be done first when there are multiple operations.

Example: ; the parentheses tell us that we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an answer of 3).

Exponents
An exponent is a superscript number placed next to another number at the top right. It indicates how many times the base number is to be multiplied by itself. Exponents provide a shorthand way to write what would be a longer mathematical expression, for example:
.

A number with an exponent of 2 is said to be “squared,” while a number with an exponent of 3 is said to be “cubed.”

The value of a number raised to an exponent is called its power. So  is read as “8 to the 4th power,” or “8 raised to the power of 4.”

The properties of exponents are as follows:
 

Property	Description
	Any number to the power of 1 is equal to itself
	The number 1 raised to any power is equal to 1
	Any number raised to the power of 0 is equal to 1
	Add exponents to multiply powers of the same base number
	Subtract exponents to divide powers of the same base number
	When a power is raised to a power, the exponents are multiplied
	Multiplication and division operations inside parentheses can be raised to a power. This is the same as each term being raised to that power.
	A negative exponent is the same as the reciprocal of a positive exponent

 
Note that exponents do not have to be integers. Fractional or decimal exponents follow all the rules above as well.

Example: .


Roots
A root, such as a square root, is another way of writing a fractional exponent. Instead of using a superscript, roots use the radical symbol () to indicate the operation. A radical will have a number underneath the bar, and may sometimes have a number in the upper left: , read as “the nth root of a.” The relationship between radical notation and exponent notation can be described by this equation: .

The two special cases of  and  are called square roots and cube roots. If there is no number to the upper left, it is understood to be a square root (). Nearly all of the roots you encounter will be square roots.

A square root is the same as a number raised to the one-half power.

When we say that a is the square root of b (), we mean that a multiplied by itself equals b: ().
A perfect square is a number that has an integer for its square root. There are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1

Order of Operations
Order of operations is a set of rules that dictates the order in which we must perform each operation in an expression so that we will evaluate it accurately. If we have an expression that includes multiple different operations, order of operations tells us which operations to do first.

The most common mnemonic for order of operations is PEMDAS, or "Please Excuse My Dear Aunt Sally." PEMDAS stands for parentheses, exponents, multiplication, division, addition, and subtraction. It is important to understand that multiplication and division have equal precedence, as do addition and subtraction, so those pairs of operations are simply worked from left to right in order.

For example, evaluating the expression  using the correct order of operations would be done like this:

P: Perform the operations inside the parentheses:
- E: Simplify the exponents. o     The equation now looks like this:
·      MD:
Perform multiplication and division from left to right: ; then o    
The equation now looks like this:

AS: Perform addition and subtraction from left to right: ; then

Subtraction with Regrouping
A great way to make use of some of the features built into the decimal system would be regrouping when attempting longform subtraction operations. When subtracting within a place value, sometimes the minuend is smaller than the subtrahend.

Regrouping enables you to ‘borrow’ a unit from a place value to the left in order to get a positive difference.

For example, consider subtracting 189 from 525 with regrouping.

First, set up the subtraction problem in vertical form:

 Notice that the numbers in the ones and tens columns of 525 are smaller than the numbers in the ones and tens columns of 189. This means you will need to use regrouping to perform subtraction:

 To subtract 9 from 5 in the ones column you will need to borrow from the 2 in the tens columns:

 
Next, to subtract 8 from 1 in the tens column you will need to borrow from the 5 in the hundreds column:

 Last, subtract the 1 from the 4 in the hundreds column:

 
Practice
P1. Demonstrate how to subtract 477 from 620 using regrouping.
P2. Simplify the following expressions with exponents:
(a)
(b)
(c)
(d)
(e)

Practice Solutions
P1. First, set up the subtraction problem in vertical form:

 To subtract 7 from 0 in the ones column you will need to borrow from the 2 in the tens column:

 
Next, to subtract 7 from the 1 that’s still in the tens column you will need to borrow from the 6 in the hundreds column:

 Lastly, subtract 4 from the 5 remaining in the hundreds column:

 
P2. Using the properties of exponents and the proper order of operations:
(a) Any number raised to the power of 0 is equal to 1:
(b) The number 1 raised to any power is equal to 1:
(c) Add exponents to multiply powers of the same base:
(d) When a power is raised to a power, the exponents are multiplied:
(e) Perform the operation inside the parentheses first: