2. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Order of operations: In mathematics order of operations is a very important process. If not used properly, an entirely different results will be obtained, which will certainly lead to errors. The following order should be maintained at all times: Parenthesis, brackets, or braces where required – always in the following order: Parenthesis – first Brackets – second Braces - third { } Braces [ ] Brackets ( ) Parenthesis For example: {2[x+5·(x-12)]+2x(3x+5)} After the parenthesis, brackets, and braces issue is understood (where the parenthesis is the most common), the order of operations is followed for the operators as well. It proceeds in the following order: Exponents such as x 2 or xy 3 Multiplication and division Addition and subtraction

3. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Order of operations summary: P Parenthesis (or brackets / braces) 1 E Exponents 2 M Multiplication or division 3 A Addition and subtraction 4 The original acronym however is typically pronounced and spelled PEMDAS where D and S represent division and subtraction. However, there is no specific order exist between multiplication and division; and no specific order exist between addition and subtraction. Please consider the following examples: 2 + 5 · 3 = 17 because no parenthesis exist in this example, the PEMDAS order is maintained: 5 is multiplied by 3 first the 2 is added. If you would add first and then multiply, the result would be entirely different. Sometimes, however, you may need to add first and then multiply. In that case parenthesis would have to applied: (2+5) · 3 = 21

4. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Additional examples: 8 + 4 / 2 = 6 Wrong answer 8 + 4 / 2 = 10 Right answer 10 – 5 · 3 = 15 Wrong answer 10 – 5 · 2 = 0 Right answer 12 ÷ 4 - 2 = 6 Wrong answer 12 ÷ 4 - 2 = 1 Right answer (12+6) · 2 = 36 Correct (10 - 2) · (5+3) = 64 Correct (6 · 4) + (5 · 3) = 39 Correct 12 + 5 - 2 · 2 – 25 · 3 = 15 Correct (5+ 3 · 2) + (8 x 4 -2) = 46 Correc t 20 + 10 ÷ 2 + 25 + (32 ÷ 4) = 48 Correct 2x + 2x · 4 = 10x Correct 4y · 4y + (16y – 6y) = 26y Correct

5. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Exponents: Exponents are smaller numbers located to the upper right corner of any number or variable. It simply indicates how many times the number is multiplied by itself and is typically called the “power” of that number. For example: 2 2 = 2 x 2 = 4 4 3 = 4 x 4 x 4 = 64 X 2 = x · x 3X 2 = 3x · 3x (5+X) 2 = (5+x) · (5+x) Exponents and order of operations: Please consider the following example: (6+4) 2 – 2 · 4 First we will do the parenthesis operation 6+4=10 Then we will square the result 10 x 10 = 100 Then we will multiply 2 and 4 = 8 And finally 100 – 8 = 92 Therefore the final result is: (6+4) 2 – 2 · 4 = 92

6. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Commutative property: Commutative property simply indicates that in multiplication and addition, the answer will be the same if you switch the places of the components. For example: 2+5 is the same as 5+2 See there’s no difference in the result, it would still be 7. Or 12 x 2 and 2 x 12 would still equal 24. Please note, however, that this would only apply to multiplication and addition. Associative property: Although similar commutative property, as it is dealing only with multiplication and addition, it applies more to the grouping of terms for easier, or more organized calculations. Typically useful in creating and applying formulas in science, engineering and architecture. Here are some examples: (6+4) + 5 = 10 + 5 =15 6 + (4+5) = 6+9 = 15 Or (2 · 3) · 5 = 30 2(3 · 5) = 30 As you can see, the grouping here is different. But the result would always be the same. In many cases the grouping of your objects, especially in longer formulas, is very important because better grouping of numbers and variables leads to better organization and reduces a chance of errors.

7. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Associative property used with variables: Associative property can also apply to variables as well as combination of numbers with variables. Here are some examples: (a + b) + c = a + (b + c) (6 + b) + a = 6 + (b + a) Or 2a · (b · c) = (2a · b) · c X · (6 · y) = (x · 6) · y Distributive property: Distributive property typically requires “opening of parenthesis” in a more innovative calculations. It presents an easy and useful method of solving more complex algebraic problems. Here are some examples: 3 · (2 + 4) First, we will multiply 3 by the first number in the parenthesis 2. Then we will multiply 3 by the second term in the parenthesis (4). Therefore the result is (3 · 2) + (3 · 4) = 6 + 12 = 18 However, if we decided to calculate in a traditional way by adding 2 and 4 within the parenthesis and then multiplying by 3 the result will still be the same 18

8. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI In many cases, in Algebra, the distributive property is used to simplify the expressions by removing the parenthesis, or sometimes to enclose certain terms within the parenthesis. Consider the following examples: Simplify this expression: 5(4 + x) 5(4 + x) = (5 · 4) + (5 · x ) = 20 + 5x This is the final answer. Sometimes you may no get a definite number, so in this case 20 + 5x is the final answer that simplifies the above expression. And as you can see the parenthesis are removed. In some cases negative signs may appear in some expressions. In that case the preceding rules will apply with regard to the negative signs. Here are some examples: -x 3 (x 2 + y 4 ) -x 3 (x 2 + y 4 ) = (-x 3 x 2 ) + (x 3 x 4 ) = (-x 6 ) + (x 7 ) = -x 6 + x 7 This is the final answer -x 6 + x 7

9. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI The Inverse property: In Algebra the inverse property comes in two forms - 1. additive inverse and 2. multiplicative inverse. For example additive inverse of any number is simply the same number with a opposite sign as in 10 and -10. If add them together you will always have a zero. So 10 + (-10) is the same as 10-10 =0 -20 + 20 = 0 -52 + 52 = 0 2a – 2a = 0 In Multiplicative inverse any number or variable has a reciprocal fraction number, which is the opposite in a fractional sense. When these numbers multiplied together the result would always be 1. For example: Number 5 can also be written as 5 1 The reciprocal (multiplicative inverse) would then be written as 1 5 3 4 4 3 · = 1 8 1 1 8 ·

10. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Some practical applications of variables: Working with variables is very useful especially in a practical life applications where letters we use represent some numbers that could apply to different measurements or technical calculations. For example: Regular house rooms come in different shapes and forms. Most commonly they appear to rectangular – having only two dimensions – the length and the width (not counting the height for this example). The typical rectangular room can be described scientifically, in the following form: Any room x y Here the length of the room is represented by the letter x and the width of the room is presented by the letter y. The perimeter could be described as: x+x+y+y or in a more traditional algebraic way as 2x + 2Y The area of the room could be described as: x multiplied by y Or simply xy

11. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI 20’ 15’ Now, if the dimensions of my room are 20’ by 15’ (which always mean the length of the room is 20 feet and the width is 15 feet), my room drawing would appear like this: Any room x y Original room drawing To calculate the perimeter of this room is quite easy by applying the previous formula 2x + 2y Therefore: (20’ · 2) + (15’ · 2) = 70’ And to calculate the area we use the formula xy 20’ · 15’ = 300’ or 300 square feet So if I wanted to carpet this room, I would need 300 sq. feet of carpeting and 70 feet of molding to nail on the floor around the room. Pretty cool, isn’t it?