 Grouping Symbols
 Multiple Grouping Symbols
 The Order of Operations
 Calculators
Inside Collection (Textbook): Derived copy of Fundamentals of Mathematics
Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:
Parentheses: ( )
Brackets: [ ]
Braces: { }
Bar:
In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.
If possible, determine the value of each of the following.
Since 3 and 8 are within parentheses, they are to be combined first.
Thus,
Since
If possible, determine the value of each of the following.
10
68
24
4
0
not possible (indeterminant)
When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.
Determine the value of each of the following.
Combine 8 and 3 first, then combine 5 and 6.
Combine 2 and 9 since they occur in the innermost set of parentheses.
Determine the value of each of the following.
54
23
4
93
74
27
Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of
Add 3 and 5, then multiply this sum by 2.
Multiply 5 and 2, then add 3 to this product.
We now have two values for one number. To determine the correct value, we must use the accepted order of operations.
Determine the value of each of the following.
Determine the value of each of the following.
33
57
0
52
3
125
7
Using a calculator is helpful for simplifying computations that involve large numbers.
Use a calculator to determine each value.
Key  Display Reads  
Perform the multiplication first.  Type  56  56 
Press  ×  56  
Type  85  85  
Now perform the addition.  Press  +  4760 
Type  9842  9842  
Press  =  14602 
The display now reads 14,602.
Key  Display Reads  
Operate inside the parentheses  Type  27  27 
Press  +  27  
Type  18  18  
Press  =  45  
Multiply by 42.  Press  ×  45 
Type  42  42  
Press  =  1890 
Place this result into memory by pressing the memory key.
Key  Display Reads  
Now operate in the other parentheses.  Type  810  810 
Press  ÷  810  
Type  18  18  
Press  =  45  
Now multiply by 105.  Press  ×  45 
Type  105  105  
Press  =  4725  
We are now ready to add these two quantities together.  Press  +  4725 
Press the memory recall key.  1890  
Press  =  6615 
Thus,
Nonscientific Calculators  
Key  Display Reads  
Type  16  16 
Press  ×  16 
Type  16  16 
Press  ×  256 
Type  16  16 
Press  ×  4096 
Type  16  16 
Press  =  65536 
Press the memory key  
Type  37  37 
Press  ×  37 
Type  37  37 
Press  ×  1396 
Type  37  37 
Press  ×  50653 
Press  +  50653 
Press memory recall key  65536  
Press  =  116189 
Calculators with 

Key  Display Reads  
Type  16  16 
Press 

16 
Type  4  4 
Press  =  4096 
Press  +  4096 
Type  37  37 
Press 

37 
Type  3  3 
Press  =  116189 
Thus,
We can certainly see that the more powerful calculator simplifies computations.
Nonscientific calculators are unable to handle calculations involving very large numbers.
Key  Display Reads  
Type  85612  85612 
Press  ×  85612 
Type  21065  21065 
Press  = 
This number is too big for the display of some calculators and we'll probably get some kind of error message. On some scientific calculators such large numbers are coped with by placing them in a form called "scientific notation." Others can do the multiplication directly. (1803416780)
Use a calculator to find each value.
13,499
6,070
1,107,495
This number is too big for a nonscientific calculator. A scientific calculator will probably give you
For the following problems, find each value. Check each result with a calculator.
26
46
1
0
3
26
97
29
1
0
90
508
19
144
1
52
25,001
14
0
152
((Reference)) The fact that 0 + any whole number = that particular whole number is an example of which property of addition?
((Reference)) Find the product.
2,690,730
((Reference)) In the statement
((Reference)) What number is the multiplicative identity?
1
((Reference)) Find the value of
"Used as supplemental materials for developmental math courses."