KFM01Sr01_Whole Numbers 11/3/2003 2:18:00 PM r10
Mathematics
Whole Numbers
Student Text
Rev 1
/ ©2003 General Physics Corporation, Elkridge, Maryland
All rights reserved. No part of this book may be reproduced in any form or by any means, without permission in writing from General Physics Corporation.
KFM01Sr01_Whole Numbers 11/3/2003 2:18:00 PM r10
TABLE OF CONTENTSFIGURES AND TABLES
OBJECTIVES
The Decimal System
Addition
Subtraction
Multiplication
Division
Other Base Number Systems
Addition of Binary Numbers
CALCULATOR OPERATIONS
Special Function Keys
Notation for Use of the Calculator
Calculator Exercises
Addition +
Subtraction
Multiplication
Division
Summary
Practice Exercises
GLOSSARY
Example Exercise answers
Practice Exercise answers
FIGURES AND TABLES
Figure 1-1 Representation of NumbersFigure 1 2 Decimal Numbers and Places
Figure 1 3 Binary Numbers and Places
Figure 1 4 Typical Basic Scientific Calculator
No Tables
OBJECTIVES
Upon completion of this chapter, the student will be able to perform the following objectives at a minimum proficiency level of 80%, unless otherwise stated, on an oral or written exam.- Without a calculator; ADD, SUBTRACT, MULTIPLY, and DIVIDE whole numbers.
- With an approved calculator; ADD, SUBTRACT, MULTIPLY, and DIVIDE whole numbers.
- Without a calculator; CONVERT between decimal and binary numbers.
MATHEMATICS - CHAPTER 1 - / 1 / © 2003 GENERAL PHYSICS CORPORATION
WHOLE NUMBERS / REV 1
KFM01Sr01_Whole Numbers 11/3/2003 2:18:00 PM r10
The Decimal System
One of man’s earliest needs was the development of a system for counting, that is, describing the number of objects in a group. We all take counting for granted, and so it is not obvious that at one time this was a difficult problem. Suppose that we have a group of objects, say rocks. How shall we describe the number of rocks in the group? The easiest method is to assign a symbol to represent the number. The figure below shows the symbols used to represent the different groups.
Figure 1-1 Representation of Numbers
The number in each group can be determined by comparison with the number of fingers on the hand, and the appropriate symbol is then assigned. The problem becomes more complicated when the number of objects exceeds the number of fingers.
What symbol should be used to represent the number of objects below?14 fourteen
Example 1 1
We could call this #!~, but this would quickly become very cumbersome. One of the great advances in mathematics was the development of the concept of place value. The same symbol can be used to represent different numbers, depending upon its position in the group of symbols that describes the number of objects. If we see the symbol 1 standing alone, we say that the group contains one object. However, if we see the symbol 10, we say the group contains ten objects. The symbol 1 then has a different meaning, depending upon its location. In this manner, a compact system for counting can be developed. A group containing two hundred and seventy-three objects can be described by the symbol 273, which means two groups of one-hundred plus seven groups of ten plus three groups of one.
There are ten separate symbols used, and these form the basis of the decimal system. These symbols are called digits, and the place values of the digits in the decimal system are multiples of ten, the base of the system.
5 4 3 2 1 / Place Title / Place ValueUnits / 1
Tens / 10
Hundreds / 100
Thousands / 1,000
Ten thousands / 10, 000
Figure 1 2 Decimal Numbers and Places
The magnitude of a number is then the sum of the digits; each multiplied by its place value.
The magnitude of 54,321 is:Digit / Place Value
5 / / 10,000 / = / 50,000
4 / / 1,000 / = / 4,000
3 / / 100 / = / 300
2 / / 10 / = / 20
1 / / 1 / = / 1
Sum / = / 54,321
Example 1 2
The magnitude of 68,095 is:Digit / Place Value
/ 10,000 / =
/ 1,000 / =
/ 100 / =
/ 10 / =
/ 1 / =
Sum / =
Example 1 3
Addition
Having developed a system for representing the number of objects in a group, or counting, it is now necessary to define rules for combining two or more groups. This process is called addition and is indicated by the plus sign (+). Suppose that we have a group of fifty-three objects represented by the decimal number 53. This means that there are five groups of ten objects plus three single objects. To this group we wish to add another group of eighteen objects represented by the decimal number 18. Thus there is one group of ten objects plus eight single objects. We have five groups of ten plus one group of ten for a total of six groups of ten. We then combine the single objects. We have three single objects plus eight single objects for a total of eleven objects. But a group of eleven objects is the same as one group of ten plus one single object. We now have a total of seven groups of ten objects plus one object, which is represented by the number 71.
This is the basic manner in which numbers are added. To shorten the process and to put it into a form that is convenient for calculation, the numbers may be tabulated vertically, taking care that the place values are properly aligned. The numbers to be added are called addends, and their total is the sum.
53addend+ 18addend
71sum
Example 1 4
Note that when we add the eight and the three, we place a 1 under the line in the units place and carry the 1 which represents the number of groups of ten over to the tens place.
1carry53addend
+ 18addend
71sum
Example 1 5
There are two laws that apply to addition. The first is the Commutative Law, which states that:
Two numbers may be added in either order and the result is the same sum.
53 + 18 = 18 + 5353 + 18 = 71
18 + 53 = 71
Example 1 6
The second law is the Associative Law, which states that:
Addends may be combined in any order and the result is the same sum.
Combine 3 + 5 + 7.3 + 5 = 8 ; 8 + 7 = 15
Or
3 + 7 = 10 ; 10 + 5 = 15
Or
5 + 7 = 12 ; 12 + 3 = 15
Example 1 7
Subtraction
The process of subtraction involves the removal of some number of objects from a group. Suppose that we have a group of 53 objects, from which we want to remove 18. The 53 objects may be divided into 5 groups of 10 plus 3 single objects. This representation, however, is not convenient for removing 1 group of 10 plus 8 single objects. Instead we divide the 53 objects into 4 groups of 10 plus 13 single objects. We now remove 1 group of 10 plus 8 single objects, leaving 3 groups of 10 plus 5 single objects. This gives us the number 35.
Subtraction is indicated by a minus sign (–). The number subtracted is called the subtrahend, the number from which the subtrahend is subtracted in called the minuend, and the result is the difference.
53minuend– 18subtrahend
35difference
Example 1 8
Notice in this example that the digit in the units place of the minuend is less than the digit in the units place of the subtrahend. Thus we need to “borrow” 10 units, or reduce the “tens” place by 1 so that we can subtract each group separately.
Borrow 10 Units53413
– 18 – 1 8
3 5
Example 1 9
The commutative law does not apply to subtraction.
53 – 18 18 – 53Example 1 10
The symbol means “unequal.”
Subtraction can be checked by adding the difference and the subtrahend. The sum will be the minuend.
Check53 minuend35 difference
– 18 subtrahend + 18 subtrahend
35 difference53 minuend
Example 1 11
Multiplication
Multiplication is a short form of addition and is indicated by the times sign (). Multiplication can also be represented by (3)(7), 37 or 3*7.
The numbers multiplied are called the multiplier and multiplicand. The result is called the product.
3multiplicand7 multiplier
21product
Example 1 12
The process of multiplication can be regarded as addition. In the above example, we could equally say that we are adding the number “3” seven times; that is, 3 + 3 + 3 + 3 + 3 + 3 + 3.
Multiplication obeys the commutative and associative laws.
Commutative Law3 7 = 7 3
Associative Law
235 = (23)5 = 2(35)=(25)3
Example 1 13
Since the commutative law applies to multiplication, the product of 3 7 can mean: (1) add the number “3” seven times or (2) add the number “7” three times. Therefore, 7 groups of 3 objects will result in a group of 21 objects, just as three groups of 7 results in a group of 21.
Since the associative law also applies to multiplication, the product of 2 3 5 can mean: (1) multiply the number “2” three times then multiply that value five times or (2) multiply the number “3” five times then multiply that value two times or (3) multiply the number “2” five times then multiply that value three times. Therefore, five groups of 6 objects will result in a single group of 30 objects, just as two groups of 15 or three groups of 10 results in a single group of 30.
The Distributive Law is applicable when numbers are multiplied and divided.
Since the distributive law applies to multiplication, the product of two times the sum of three plus five mean: (1) add the numbers “3” and “5” then multiple the sum two times or (2) distribute the multiplier “2” into the values to be summed. That means multiply each of the numbers to be added by “2” then add those values together; multiply 2 times 3 and add that quantity to the product of 2 times 5. In both cases the answer is 16. Therefore, two groups of 8 objects will result in a single group of 16 objects, just as two groups of 3 objects plus two groups of 5 objects results in a single group of 16.
Distributive Law2 (3 + 5) = 2 (8) = 16
2 (3) + 2 (5) = 6 + 10 = 16
Example 1 14
Division
Division is indicated by the division sign (). Division can also be represented by 3/5, , or . Placing one number over another separated by a horizontal line also indicates division.
Example 1 15
The number that is to be divided is called the dividend; the number that is divided into the dividend is the divisor. The result is called the quotient.
28 4 = 7Dividend Divisor = Quotient
Example 1 16
Multiplying the divisor and quotient can check division. The result will be the dividend.
28 4 = 7Dividend Divisor = Quotient
Check:
4 7 = 28
Divisor Quotient = Dividend
Example 1 17
Division is really a short form of subtraction. In the above example we start with a group of 28 objects, and remove 4 objects from the group at a time. After this has been done 7 times, we will have depleted the group. That is, none will remain.
Suppose instead that we started with a group of 29 objects. If we now remove 4 objects at a time, we see that when we have done this 7 times there will be 1 object left. This is called the remainder, sometimes abbreviated R or r.
Quotient / 7 / r1 / RemainderDivisor / / Dividend
Example 1 18
To check a division problem with a remainder, multiply the divisor by the quotient and add the remainder to obtain the dividend.
29 4 = 7 r1Dividend Divisor = Quotient + Remainder
Check:
4 7 = 28
Divisor Quotient = Check number
28 + 1 = 29
Check number + Remainder = Dividend
The answer checks.
Example 1 19
The Distributive Law is applicable when numbers are multiplied and divided.
Since the distributive law applies to division, the sum of eight plus twelve divided by 4 mean: (1) add the numbers “8” and “12” then divide the sum by four or (2) distribute the divisor “4” into the values to be summed. That means divide each of the numbers to be added by “4” then add those values together; divide 8 by 4 and add that quantity to the quotient of 12 divided by 4. In both cases the answer is 5.
Distributive Law(8 + 12) 4 = (20) 4 = 5
(8) 4 + (12) 4 = 2 + 3 = 5
Example 1 20
Other Base Number Systems
The decimal or base ten system probably arose as a result of the fact that there are ten fingers on the hand. There are other systems, however, which may be more suitable depending upon the application. For example, a modern digital computer is really nothing more than a complex series of switches. Since there are only two positions of a switch, (i.e. open or closed), it is more convenient for a computer to operate in a number system having only two digits. Such a system is called a binary system, and contains only the digits 0 and 1. This is a base 2 number system, and the place values are multiples of two. It is similar to the decimal system where the place values are multiples of ten.
1 0 1 1 0 / Place Value1
2
4
8
16
Figure 1 3 Binary Numbers and Places
The decimal system equivalent of a binary number is the sum of each digit multiplied by its place value.
The binary number 10110 is written 101102. This shows the base is two, so that the number is not interpreted as ten thousand one hundred and ten. Strictly speaking, we should write the decimal number 22 as 2210, but since most of our work is in the decimal system, the base is understood to be ten and the subscript is omitted.
The decimal equivalent of 10110 is:Digit / Place Value
1 / / 16 / = / 16
0 / / 8 / = / 0
1 / / 4 / = / 4
1 / / 2 / = / 2
0 / / 1 / = / 0
Sum / = / 22
Example 1 21
Therefore, the binary number 10110 represents the same number of objects as does the decimal number 22. In order to prevent confusion a subscript that indicates the base is usually placed after the last digit.
The decimal equivalent of 10001 is:Digit / Place Value
/ 16 / =
/ 8 / =
/ 4 / =
/ 2 / =
/ 1 / =
Sum / =
Example 1 22
Addition of Binary Numbers
Thus far we have only added numbers in the base ten system, but addition can be performed in any number system. The rules are exactly the same. Suppose we wish to add the binary numbers 1002 and 1112. The binary number 1002 represents one group of four objects plus no groups of two objects plus no single objects. The number 1112 represents one group of four objects plus one group of two objects plus one single object. We add the groups separately to obtain two groups of four objects plus one group of two objects plus one single object. However, two groups of four objects are the same as one group of eight objects. Thus the total is one group of eight objects plus one group of two objects, plus one single object, written as 10112.
1112+ 1002
10112
Example 1 23
Notice that we have “carried” the unit in the “fours” place over to the “eighths” place, just as we did when we added numbers in the decimal system. The addition can be checked by converting the numbers to the decimal system.
Add 1112 + 10021112
+ 1002
???
02 +12 = 12
1112
+ 1002
??12
12 + 02 = 12
1112
+ 1002
?112
12 + 12 = 102
1112
+ 1002
10112
(Cont'd in next Column)
To check the answer convert the two addends to decimal numbers and add them in decimal numbers. Then convert the binary sum to a decimal number and compare it to the decimal sum.
1112
DigitPlace Value
14=4
12=2
11= 1
Sum=7
1002
DigitPlace Value
14=4
02=0
01= 0
Sum=4
Sum = 7 + 4 = 11
10112
DigitPlace Value
18=8
04=0
12=2
11=1
Sum=11
The two sums agree.
Example 1 24
CALCULATOR OPERATIONS
Many calculators are available. Each may be a little different. For the purpose of this discussion, a basic scientific calculator will be needed. Most calculators work on the same principles. Each student should refer to the reference manual for his or her calculator. This discussion does not address calculators that operate on a programming principle.
The following section reviews the general use of common function keys on a scientific calculator. Individual chapters in this course address the application of special calculator functions. Those chapters discuss the mathematical principles and are followed by calculator exercise sections providing guidance and practice on those applicable functions of the module.
Figure 1 4 Typical Basic Scientific Calculator
Special Function Keys
CLEAR ENTRY/CLEAR KEY CE/C AC
The CE/C key clears the last operation and the display. Press the key twice to clear all operations except the memory. To clear the memory, press CE/C then STO.
Note:Some calculators break this function into two separate keys. Usually labeled "clear (C)" and "all clear (AC)". The "clear" key clears the last entry and the "all clear" key clears the display and all pending operations.
MEMORY KEY STO
The STO key allows a number to be stored into memory for reuse. This feature is useful if a value is to be used repeatedly. It also allows one calculation to be performed, the answer stored into memory for use later in another calculation.
Press the STO key (store to memory) to enter the displayed number into memory. Any number already in memory will be overwritten.
Most calculators have at least one memory. Some calculators have additional memory functions, refer to your reference manual on how use additional memory functions. Typically calculators with more than one memory will require a number to be entered with the STO key. For example, STO 01 means store the displayed number in memory 01; STO 20 means store the number in memory 20.
MEMORY RECALL KEY RCL
The RCL key allows a number stored in memory to be recalled for use.
Press the RCL key (recall from memory) to retrieve the number in memory and display it. The number remains in the memory. This allows the number to be used again. Pressing RCL overwrites any number on the calculator display.
Typically calculators with more than one memory will require a number to be entered with the RCL key. RCL 01 means recall the number stored in the 01 memory. RCL 20 means recall the number stored in memory 20.
CONSTANT KEY K
Some calculations often contain repetitive operations and numbers. The K, constant, is a timesaving function that allows a single keystroke to perform a single operation on the displayed number.
For example, if a large number of operations require a number to be multiplied by 6.02 × 1023, the K key can be used. Enter 6.02 × 1023, then press the times key, then the K key; this "programs" the calculator for the required operation. To use this feature enter the number to be multiplied by the constant and press the K key. The calculator will automatically multiply the displayed number by 6.02 × 1023, saving you keystrokes. Pressing the Clear or All Clear key removes the number from the K memory.
MEMORY EXCHANGE KEY EXC
The EXC key, swaps the displayed number with the number in memory.
RECIPROCAL KEY 1/x
When pressed, it divides the displayed number into one.
Notation for Use of the Calculator
Throughout this textbook after discussing the theory of a mathematical operation and showing examples to solve problems using the theory, examples will show how to solve the problem using a calculator. The format is a three column graphic. The first column represents the initial number to be entered. The second column represents calculator key(s) used to perform the mathematical operation desired. The third column represents the display shown on the calculator, this allows the users to ensure they are following the required manipulations of the calculator.