1. Algebraic Expressions Applications in atomic science

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

4. 1 3 -9 -27 3 = = -27 3 (-27) ( ) = d = 1 2 ) ( b d 1 2 ) ( b 2 b 1 ) (1)
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.
A common symbol technicians, scientists and engineers use for multiplication.
TWO EXAMPLES
The fraction one third is the reciprocal of 3
(a) Specific Situation 1 3 -9
-27 3 = =
-27 3
(-27) ( ) =
Dividing –27 by 3 is the same as multiplying –27 by the reciprocal of 3.
the fraction 1 ) ( b
is the reciprocal of b
(b) General Situation d = 1 2 ) ( b d 1 2 ) ( b 2 b 1 ) ( d d 2 ) ( b = ( =

5. 4 = 1 1 1 = 4 0.25 1 4 4 EASY PRACTICE PROBLEMS
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.
4 EASY PRACTICE PROBLEMS
The fraction one-fourth
(a) Divide the number 1 by
the number 4 4 = 1 ? 1 1 = 4
0.25 1
is the reciprocal of the number 4 1 4
Dividing 1 by 4 is the same as multiplying 1 by the fraction
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

6. 9 = 1 0.11 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.
4 EASY PRACTICE PROBLEMS (continued)
(b) 9 = 1 ?
0.11 1 = 9

7. 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.
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? d = 1 10 ) ( b ( d 1 10 ) b = = 5 1 10 ) ( 36 ? 1 10 ) ( 5 36 = 1 b
The variable b
and the variable
are reciprocals of each other. =
(0.10)
(0.14) =
0.014

8. l l l ) ( 10 b d 1 = = ? (d) 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.
4 EASY PRACTICE PROBLEMS (continued)
(d)
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. l
the Greek letter Lambda 10 b ) ( d 1 l = = ? l
when b equals 36 and d equals 5.

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

10. Evaluating Expressions
Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebraic symbols.
The previous example problem used the Greek letter lambda as well as the letters “d” and “b”. = l 10 b ) ( d 1
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. 10 b ) ( 1 l 1 = d 1

11. l l l l ) ( ( ) ( ) ( ) 10 d 1 1 1 = d 1 10 b 1 1 ( 10 ) 1 b d = = 5
Evaluating Expressions
What is the reciprocal of ? 10 d ) ( 1
Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebra symbols. 1 l 1 ( ) = d 1 10 b 1
when b1 equals 36 and d1 equals 5. l 1 ( 10 ) 1 b d = = 5 10 ( ) 36 =
1800 ?
PRACTICE PROBLEM
when b1 equals 36 and d1 equals 5. d1 10 1 ( ) b1 l 2 = d1 10 ) b1 ( = ? l 2 5 10 1 ( ) 36 50 =
0.72 = = 36

12. Evaluating Expressions
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.
substitute different numbers.
2 EASY EXAMPLE PROBLEMS
These examples use the following algebraic expression; b 1 = 10 ) ( d l
(1)
Let b1 equal 36 and d1 equal 5. = ? ( 10 ) 1 b d 5 36 l
1800
(2)
This time, let b1 equal 35 and d1 equal 5. l 2 = ( 10 ) 1 b d = 5 10 ( ) 35 =
1750 ?

13. l l ( ) ( ) ( ) ( ) ( ) ( ) 1 = 50 35 = 10 7 = 1 10 d b = ? 1.43 1 =
Evaluating Expressions
Sometimes an engineer, scientist or technician may select symbols that are similar when the algebraic expressions are different.
This often happens when there is a connection between the answers after the expressions have been evaluated.
2 EASY PRACTICE PROBLEMS
In both problems b1 equals 35 and d1 equals 5.
(1) 1 = ( 50 ) 35 = ( 10 ) 7 = l 1 10 ) ( d b
= ?
1.43 1 = ( 35 ) 50 = ( 7 ) 10
(2) = l 2 10 ) ( d 1 b
= ?
0.7

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

15. Evaluating Expressions
3 quick review questions to see what we remember. l 1 l 2 If
= 1.43
and
= 0.7 l 2 1
What is the connection between
and ? 1)
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.
Try this with a calculator. Is there a problem?
What is the reciprocal of ? 10 d ) ( 1 3) d 1 10 ) (

16. Is one-half the reciprocal of 2? Why/why not?
Evaluating Expressions
WHAT DO YOU THINK? 1)
Is one-half the reciprocal of 2? Why/why not? 2)
Do all fractions have reciprocals? 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?