1. Algebraic Expressions 1 Applications in atomic science

2. Algebraic Expressions 2 Scientists, engineers and technicians need, develop, and use mathematics to explain, describe, and predict what nature, processes, and equipment do. Many times the math they use is the math that is taught in algebra 1!

3. Algebraic Expressions 3 The Objective of this presentation is to show: how to evaluate algebraic expressions involving multiplication and division of real numbers.

4. Algebraic Expressions 4 1) The rules for dividing real numbers involve the mathematical concept of reciprocals. TWO EXAMPLES (-27)= 1 3 The fraction one third is the reciprocal of 3 Dividing –27 by 3 is the same as multiplying –27 by the reciprocal of 3. d= 1 2 )(b == d 1 2 )()( b 1 ( 2b1 ))()( 11dd 2 )( b (a) Specific Situation -9 -27 3 = = 3 The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. Evaluating Expressions () (b) General Situation the fraction 1 )( b is the reciprocal of b A common symbol technicians, scientists and engineers use for multiplication.

5. Algebraic Expressions 5 Evaluating Expressions The rules for dividing real numbers involve the mathematical concept of reciprocals. 4 EASY PRACTICE PROBLEMS The fraction one-fourth Dividing 1 by 4 is the same as multiplying 1 by the fraction 4 = 1 ? 0.25 1= 1 4 1 1 is the reciprocal of the number 4 1 4 Dividing 1 by 4 is the same as multiply 1 by the reciprocal of the number 4 1 4 is the reciprocal of the number 4 (a) Divide the number 1 by the number 4 The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

6. Algebraic Expressions 6 9 = 1 ? 0.111= 1 9 (b) 4 EASY PRACTICE PROBLEMS (continued) Evaluating Expressions The rules for dividing real numbers involve the mathematical concept of reciprocals. The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

7. Algebraic Expressions 7 4 EASY PRACTICE PROBLEMS (continued) (c) If the values of d and b are 5 and 36 respectively, what is the value of the following algebraic expression? = (0.10) (0.14) 1 10 )( 5 36 )( ? =5 1 10 )()( 36 1 (d 1 10 ))( b 1 = = 0.014 = d= 1 10 )(b )( 1 b are reciprocals of each other. and the variable The variable b Evaluating Expressions The rules for dividing real numbers involve the mathematical concept of reciprocals. The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

8. Algebraic Expressions 8 Sometimes it is fun in Algebra to use a letter from the Greek alphabet as well as letters like “d” and “b”. Try the following problem using the Greek letter Lambda. (d) the Greek letter Lambda = ? = when b equals 36 and d equals 5. 10b )( d 1 4 EASY PRACTICE PROBLEMS (continued) Evaluating Expressions The rules for dividing real numbers involve the mathematical concept of reciprocals. The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

9. Algebraic Expressions 9 4 Easy Practice Problems (continued) (d) = = 10b )( d 1 5 ()36 =5360() 1800 = ? (d) 10 )( b 1 the Greek letter Lambda = ? = 10b )( d 1 What is the reciprocal of ? 10b )( 1 Evaluating Expressions

10. Algebraic Expressions 10 Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebraic symbols. = 10b )( d 1 The previous example problem used the Greek letter lambda as well as the letters “d” and “b”. A technician might see this algebraic expression with lambda and the letters “d” and “b” replaced by symbols that are combinations of letters and numbers. =d 1 10b )( 1 1  Evaluating Expressions

11. Algebraic Expressions 11 Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebra symbols. = 10 )( 1 d 1 b 1 when b 1 equals 36 and d 1 equals 5. = ? = ( 10 ) )( 1 b 1 d 1 5 ()36   =1800 PRACTICE PROBLEM  == = d1d1 10)b1b1 ( = ?  when b 1 equals 36 and d 1 equals 5. d1d1 10 1 () b1b1 36 510 1 () 36 50 =0.72 What is the reciprocal of ? 10d )( 1 Evaluating Expressions

12. Algebraic Expressions 12 Scientists, technicians and engineers also use algebraic symbols that are combinations of letters and numbers because they often work with the same algebraic expression but substitute different numbers. b 1 = 10 )( 1 d 1 = ? = ( 10 ) )( 1 b 1 d 1 5 ()36  =1800  = Let b 1 equal 36 and d 1 equal 5. 2 EASY EXAMPLE PROBLEMS (1) (2) These examples use the following algebraic expression; This time, let b 1 equal 35 and d 1 equal 5. ? = ( 10 ) )( 1 b 1 d 1 5 ()35=1750 substitute different numbers. Evaluating Expressions

13. Algebraic Expressions 13 = ? (1) (2) 0.7 Sometimes an engineer, scientist or technician may select symbols that are similar when the algebraic expressions are different. 2 EASY PRACTICE PROBLEMS In both problems b 1 equals 35 and d 1 equals 5. This often happens when there is a connection between the answers after the expressions have been evaluated. =  10 ) ( d 1 b 1 1.43 = ( 10 ) 7 1 = ( 50( ) 35 ) =  10 ) ( d 1 b 1 = ? 1 = ( 35( ) 50 )= ( 7 ) 10 Evaluating Expressions

14. Algebraic Expressions 14 = ? (1) (2) 0.7 =  10 ) ( d 1 b 1 1.43= ( 10 ) 7 1 = ( 50( ) 35 ) =  10 ) ( d 1 b 1 = ? 1 = ( 35( ) 50 )= ( 7 ) 10   ()() = (1.43) (0.7) = 1.00 What is the connection between and ?     is the reciprocal of.   is the reciprocal of. or, if you wish 1 Evaluating Expressions

15. Algebraic Expressions 15 3 quick review questions to see what we remember. 1)   If = 1.43 and = 0.7   What is the connection betweenand? One is the reciprocal of the other. 2)What is the answer if you multiply reciprocals together? You always get the number 1 as the answer. 3) Try this with a calculator. Is there a problem? What is the reciprocal of ? 10d )( 1 d 1 )( 1 Evaluating Expressions

16. Algebraic Expressions 16 WHAT DO YOU THINK? 1) 2)Do all fractions have reciprocals? Why/Why not? Is one-half the reciprocal of 2? Why/why not? 3)Two of the most popular manufacturers of calculators (TI and HP) have a different style (ways to do calculations) for getting answers to multiplication and division problems. One of them was developed with a knowledge (use) of reciprocal in mind. Which one is it? Why? 4)Use the Web (if you have to) or a real slide rule if you have one, and examine the arrangement of the scales on a slide rule. One of the scales is know as the reciprocal scale. Which one is it and why is it named so? Evaluating Expressions

17. Algebraic Expressions 17