algebraic expressions 1 applications in atomic science

Algebraic Expressions

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Algebraic ExpressionsApplications in atomic science


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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!


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The Objective of this presentation is to show:

how to evaluate algebraic expressions involving multiplication and division of real numbers.


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1)

The rules for dividing real numbers involve the mathematical concept of reciprocals.

TWO EXAMPLES

(-27)

=13

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 =12

)( b

= =

d12

)( )(b1

( 2

b1) )()(1 1d d2

)( b

(a) Specific Situation

-9-273

==-27 3

The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

Evaluating Expressions

( )

(b) General Situationthe fraction

1

)(

bis the reciprocal of b

A common symbol technicians, scientists and engineers use for multiplication.


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4 EASY PRACTICE PROBLEMSThe fraction one-fourth

Dividing 1 by 4 is the same as multiplying 1 by the fraction

4

=1 ? 0.25

1 =14

1 1

is the reciprocal of the number 4

14 Dividing 1 by 4 is the same as multiply 1 by the

reciprocal of the number 4

14

is the reciprocal of the number 4

(a) Divide the number 1 by the number 4



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9

=1 ? 0.111 =19

(b)

4 EASY PRACTICE PROBLEMS (continued)





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(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)

110)( 5

36)(

? =5110 )( )(36

1( d110 ) )( b

1 =

=

0.014=

d =110 )( b) (

1b

are reciprocals of each other.

and the variable

The variable b





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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.

10 b)(d

1






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4 Easy Practice Problems (continued)

(d)

=

=10 b

)(d1

5 10 ( )36

= 5 360 ( )1800 =

?(d) 10 )( b1

the Greek letter Lambda

= ?= 10 b

)(d1

What is the reciprocal of ? 10 b

)(1



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Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebraic symbols.

= 10 b)(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.

= d1 10 b)(

1

1



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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

d1 b1 when b1 equals 36 and d1 equals 5.

= ? =( 10 ) )(1

b1d1 5 10 ( )36

=1800

PRACTICE PROBLEM

= =

=

d110 )b1 (

=

?

when b1 equals 36 and d1 equals 5.

d110 1

( )b1

36 510 1

( )36

50=0.7

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What is the reciprocal of ? 10 d )( 1



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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.

b1=

10 )(

1d1

= ? =( 10 ) )(1

b1d1 5 10 ( )36=1800

=

Let b1 equal 36 and d1 equal 5.

2 EASY EXAMPLE PROBLEMS

(1)

(2)

These examples use the following algebraic expression;

This time, let b1 equal 35 and d1 equal 5.

? =( 10 ) )(1

b1d1 5 10 ( )35 =1750

substitute different numbers.



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 b1 equals 35 and d1 equals 5.

This often happens when there is a connection between the answers after the expressions have been evaluated.

= 10 )( d1b1 1.4

3 =(10 )

7

1=(50( )

35)

= 10 )( d1b1= ?

1=(35( )

50) =( 7 )

10



14

= ?(1)

(2)0.7

= 10 )( d1b1 1.4

3 =(10 )

7

1=(50( )

35)

= 10 )( d1b1= ?

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



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3 quick review questions to see what we remember.

1)

If = 1.43 and = 0.7

What is the connection between and ?

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 ? 10 d )( 1

d110 )(

1



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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?



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algebraic expressions 1 applications in atomic science

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