Multiplying and Dividing Real Numbers

 

Multiplicative Inverse (or reciprocal) For each real number a, except 0, there is a unique real number  such that

 
 

In other words, when you multiply a number by its multiplicative inverse the result is 1.  A more common term used to indicate a  multiplicative inverse is the reciprocal.  A multiplicative inverse or reciprocal of a real number a (except 0) is found by “flipping” a upside down.  The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.

 
 

Example 1:   Write the reciprocal (or multiplicative inverse) of -3.

 

The reciprocal of -3 is -1/3, since -3(-1/3) = 1. When you take the reciprocal, the sign of the original number stays intact.  Remember that you need a number that when you multiply times the given number you get 1.  If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no.

 
 

Example 2:   Write the reciprocal (or multiplicative inverse) of 1/5.

 

The reciprocal of 1/5 is 5, since 5(1/5) = 1.

 
 
 
 

Quotient of Real Numbers If a and b are real numbers and  b is not 0, then

 
 
 
 

Multiplying or Dividing Real Numbers

 

Since dividing is the same as multiplying by the reciprocal, dividing and multiplying have the same sign rules.  Step 1:   Multiply or divide their absolute values.    Step 2:   Put the correct sign.   If the two numbers have the same sign, the product or quotient is positive.  If they have opposite signs, the product or quotient is negative.

 
 

Example 3:  Find the product  (-4)(3).

 

(-4)(3) = -12.  The product of the absolute values 4 x 3 is 12 and they have opposite signs, so our answer is -12.

 
 
 

Example 4:  Find the product .

 

*Mult. num. together *Mult. den. together *(-)(-) = (+) *Reduce fraction

 

The product of the absolute values 2/3 x 9/10 is 18/30 = 3/5 and they have the same sign, so that is how we get the answer 3/5. Note that if you need help on fractions go to Tutorial 3: Fractions

 
 
 
 

Example 5:  Find the product

 

Working this problem left to right we get:

 

*(3)(-2) = -6 *(-6)(-10) = 60

 
 
 

Example 6:  Divide   (-10)/(-2).

 

(-10)/(-2) = 5  The quotient of the absolute values 10/2 is 5 and they have the same signs, so our answer is 5.

 
 
 

Example 7:  Divide .

 

*Div. is the same as mult. by reciprocal *Mult. num. together *Mult. den. together *(+)(-) = - *Reduce fraction

 

The quotient of the absolute values 4/5 and 8 is 4/40 = 1/10 and they have opposite signs, so our answer is -1/10.   Note that if you need help on fractions go to Tutorial 3: Fractions

 
 

Multiplying by and  Dividing into Zero a(0) = 0 and 0/a = 0   (when a does not equal 0)

 

In other words, zero (0) times any real number is zero (0) and zero (0) divided by any real number other than zero (0) is zero (0).

 

Example 8:   Multiply  0(½).

 

0(½) = 0. Multiplying any expression by 0 results in an answer of 0.

 
 
 
 

Example 9:   Divide 0/5.

 

0/5 = 0. Dividing 0 by any expression other than 0 results in an answer of 0.

 
 
 

Dividing by Zero a/0 is undefined

 

Zero (0) does not go into any number, so whenever you are dividing by zero (0) your answer is undefined.    Example 10:   Divide 5/0.

 

5/0 = undefined.  Dividing by 0 results in an undefined answer.

 
 

Example 11:   Simplify .

 

Since we have several operations going on in this problem, we will have to use the order of operations to make sure that we get the correct answer.  If you need to review the order of operations go to Tutorial 4: Operations of Real Numbers.

 

*Evaluate inside the absolute values   *Subtract    *(-)/(-) = +

 

Example 12:   Evaluate the expression    if  x = -2 and y = - 4.

 

To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations. Plugging -2 for x and - 4 for y and simplifying we get:

 

*Plug in -2 for x and -4 for y *Exponent *Multiply *Add

Addition and Subtraction of Real Numbers

All the basic operations of arithmetic can be defined in terms of addition, so we will take it as understood that you have a concept of what addition means, at least when we are talking about positive numbers.

Addition on the Number Line

A positive number represents a distance to the right on the number line, starting from zero (zero is also called the origin since it is the starting point). When we add another positive number, we visualize it as taking another step to the right by that amount. For example, we all know that 2 + 3 = 5. On the number line we would imagine that we start at zero, take two steps to the right, and then take three more steps to the right, which causes us to land on positive 5.

Addition of Negative Numbers

What does it mean to add negative numbers? We view a negative number as a displacement to the left on the number line, so we follow the same procedure as before but when we add a negative number we take that many steps to the left instead of the right.

Examples: 

2 + (–3) = –1

First we move two steps to the right, and then three steps to the left:

(–2) + 3 = 1

We move two steps to the left, and then three steps to the right:

 (–2) + (–3) = –5

Two steps to the left, and then three more steps to the left:

From these examples, we can make the following observations:

1.      If we add two positive numbers together, the result will be positive

2.      If we add two negative numbers together, the result will be negative

3.      If we add a positive and a negative number together, the result could be positive or negative, depending on which number represents the biggest step.

 Subtraction

There are two ways to define subtraction: by a related addition statement, or as adding the opposite.

Subtraction as Related Addition

a – b = c if and only if a = b + c

Subtraction as Adding the Opposite

For every real number b there exists its opposite –b, and we can define subtraction as adding the opposite:

a – b = a + (-b)

·        In algebra it usually best to always think of subtraction as adding the opposite

Distinction Between Subtraction and Negation

The symbol “–” means two different things in math. If it is between two numbers it means subtraction, but if it is in front of one number it means the opposite (or negative) of that number.

Subtraction is binary (acts on two numbers), but negation is unary (acts on only one number).

          Calculators have two different keys to perform these functions. The key with a plain minus sign is only for subtraction:

Negation is performed by a key that looks like one of these:

Remember that subtraction can always be thought of as adding the opposite. In fact, we could get along just fine without ever using subtraction.

Subtraction on the Number Line

Addition of a positive number moves to the right, and adding a negative moves to the left.

Subtraction is just the opposite: Subtraction of a positive number moves to the left, and subtracting a negative moves to the right.

·     Notice that subtracting a negative is the same thing as adding a positive.

Variable Expressions

The principle mathematical innovation of Algebra (as opposed to Arithmetic) is the use of letters, as variables, for unknown numbers. Arithmetic is the mathematics of the basic operations (add, subtract, multiply, divide), but always as applied to known, specific numbers. With Algebra, we are freed to use a letter to represent any possible number. It is important to remember at all times that the letter variable simply holds the place of a number, and any operations we apply to it must follow the Arithmetic laws of real numbers.
The first common task you will need to master is to substitute given values for specific variables. This is the task of turning an unknown letter into a known number. To do so, replace the letter variable by empty parentheses ( ) to preserve the place of the variable. Then drop the known value into the parentheses. Follow the usual Arithmetic rules (order of operations) to simplify.
Whenever you see a statement like "x = a number," you are being told exactly what x is. Knowing the value for x, you are free to replace the x in that expression by its actual value, for they are the same.
Example 1 Evaluate 3x - 4 when x = 5.
Solution: 3x - 4 means 3 times a number, then subtract 4.

Write:	3( ) - 4	(The empty parentheses hold the space of the x)
	= 3(5) - 4	(Fill the parentheses with the given value of x)
	= 15 - 4	(Simplify according to the Order of Operations)
	= 11

Example 2 Evaluate 2ab - 5b when a = -2 and b = 3.
Solution: Write 2( )( ) - 5( ) where the parentheses save spaces for a and b.
= 2( -2)( 3 ) - 5( 3 )
= -12 - 15 (Do the multiplications, then add)
= -27
Exercises
If x = -2, y = 4, and a = 1, evaluate each of the following:

1) 4x - 3	2) y - x	3) axy
4) 3x	5) a + y	6) x - 2a + y
7) 2a - 5x + y	8) a - xy	9) 3 - 2x

Evaluate the following for a = -1, b = -2, and c = 5:

10) a + 5	11) a + b - c	12) 4a - 2bc
13) c - 3a	14) c - b	15) (3c + a)/b
16) c(3a - 2b)	17) c - 4(b + 2)	18) (a - 3) (2c + 1)

Many times in Algebra, you will be working with variable expressions without being given any values to fill in for the variables. In this case, you must be especially careful to preserve the meaning of the expression in any work. You will be asked to simplify expressions, which means to rewrite the expression in a better form. Usually, the "better form" involves fewer symbols, but it must have the same meaning for all possible values of the variables.
The most important idea for working with variable expressions is to work only with the separate terms. A term will consist of a signed number multiplied by one or more variables. In a variable expression, you may have several terms. Each + or - sign will signal the start of a new term. For example, the variable expression 5x - 3y + 4ab consists of exactly three terms: +5x and -3y and +4ab. Terms may be moved within a variable expression, but be sure to carry the sign with each term. 5x + 4ab - 3y is the same expression, as is -3y + 4ab + 5x.
For now, you will need to be able to simplify additions and subtractions and a few simple multiplications. There is just one simple rule: Add or subtract like terms only. Like terms must have the same variable part.
Example 3 Simplify 3x + 7 + 2x - 3.
Solution: The terms can be reordered to group the x-terms and constant terms (terms with no variable).

3x + 7 + 2x - 3	(Arrange the 3x and 2x terms together. Put the
= 3x + 2x + 7 - 3	+7 and -3 together)
= 5x + 4	(Add 3x and 2x to get 5x; Subtract +7 - 3 to get +4)

Example 4 Simplify 5x + 1 - 6x - 4.

Solution:	5x + 1 - 6x - 4
	= 5x - 6x + 1 - 4
	= -1x + -3
	= -x - 3

(Note that 1x can be written as simply x. This is the same as saying "a box" rather than "one box". When a variable term has no number in front, you may assume that there is really a "1" there. So x means 1x and -x means -1x.)
Example 5 Simplify 2(5x - 8).

Solution:	2(5x - 8)
	= 2(5x) - 2(8)	2 times the group means 2 times each term.
	= 10x - 16

Example 6 Simplify -3(y - 4).

Solution:	-3(y - 4)
	= -3(y) - 3(-4)	Multiply each term by -3.
	= -3y + 12	Note that - times - is +.

More Exercises
Simplify (add like terms only):

19) 4x + 3x	20) 2x - 7x	21) 10y + 2y - 5y
22) 4x - 2 - 3x	23) 6c - 8 + c + 3	24) -x - 2y + 3x
25) 5x + 7 - 4x - 10	26) a - 3b + 10a + 5b - 2	27) 2x - 4y - 2x + 7y

Simplify (multiply by each term in the parentheses)

28) 2(3x + 1)	29) 3(5x - 8)	30) 7(2y - 3)
31) 4(8a + 3)	32) -2(3c + 2)	33) -5(2x + 4)
34) 2(9x - 4)	35) -(x + 3)	36) -(2y - 3)
37) 5(3x + 2y - 5)	38) -3(2x - 5y + 6)	39) -(10 - 5x)

Simplify (Multiply through parentheses first, then add the like terms):

40) 2x + 3(4x - 3)	41) 9y + 2(3y + 1)	42) 4 + 3(5x - 4)
43) 3a - 2(5a + 1)	44) 6 - (x - 5)	45) 2(3x - 2) + 4

(Bonus Round) Simplify:

46) 4y - 3 when y = x + 2	47) 2z + 6 when z = 3x + 1
48) 1 - 2a when a = 4b + 3	49) 3 - y when y = 2x - 9

Formula Problems
Sometimes a word problem comes with a formula. Such problems appear at all levels of mathematics learning, from beginning algebra to calculus, and beyond. When given a formula, it is important to come to an understanding of what kind of number each letter in the formula represents. If you can match every number given in the word problem with a letter in the formula, you will be able to solve the problem.
Use the following steps to approach a formula problem.
1. Identify and write down the appropriate formula. Make a list showing what each letter in the formula stands for.
2. Match each number in the problem statement with one of the letters in your list. Fill these values into the corresponding spaces in the formula.
3. Perform the indicated arithmetic operations to find the answer to the problem. Example 7 Given a Celsius temperature C, the Fahrenheit temperature is found by the formula . What is the Fahrenheit temperature corresponding to 20° C?
Solution: Use , where C = Celsius temp. = 20°
So = 36 + 32 = 68° F.
Example 8 The amount of simple interest paid on a savings account is I = Prt, where P is the amount of the deposit, r is the interest rate, and t is the time (in years) that the deposit is kept in the account. How much interest is paid on a $4000 deposit left for 5 years in an account earning 6% simple interest?
Solution: Use the formula I = Prt.

The known quantities are:	P = deposit = $4000
	r = rate = 6% = 0.06
	t = time = 5 years

So I = Prt = (4000)(0.06)(5) = $1200.

More Exercises
Choose a formula from the list of frequently used formulas and use it to solve the problem.

50. What is the sale price of a pair of $55 boots marked 30% off?
51. How much interest is paid on an account earning 4% simple interest if $1200 is left on deposit for 8 years?
52. How far does a car travel in 3 and a half hours at 45 miles per hour?
53. If the temperature is 59° F, what is the Celsius temperature?
54. Find the area and the perimeter of a rectangle that is 14 centimeters long by 17 centimeters wide.
55. Find the area of a triangle with a base of 42 feet and a height of 15 feet.
56. An object is thrown straight up from a height of 4 feet above the ground at 75 feet per second. Find its height after 2 seconds.