Basic Arithmetic Operations Learning Outcome When you complete this module you will be able to: Perform basic arithmetic operations without the use of a calculator. Learning Objectives Here is what you will be able to do when you complete each objective: 1. Add and subtract integers. 2. Multiply and divide whole and decimal numbers. 3. Evaluate equations involving combinations of addition, subtraction, multiplication and division, and powers in the proper sequence. 1
INTRODUCTION While much of this material may be very elementary to some students, a review may refresh the memory and sharpen the skills of those students who finished their schooling many years ago. If you think your mathematical skills are sharp, then skip to the self-tests at the end of this module and try them. If you have trouble with some questions, then find the corresponding part of the module and study it carefully. Because many people now rely on calculators for doing even the most basic arithmetic operations, they have forgotten many of the skills they learned in school. However, in order to manipulate algebraic expressions and solve equations, the student must understand how to do the basic arithmetic operations without a calculator. Therefore, in studying this module, students are urged to set their calculator aside and to do the calculations by hand unless specific instructions to the contrary are given. BASIC NUMBERS When working with numbers, the main objectives are speed and accuracy. Most people now rely on the electronic calculator to accomplish both objectives. However, unless a person understands how to manually perform basic mathematical operations, the accuracy of answers given by the calculator cannot be assessed. Some basic terms require definition at the outset. Natural numbers are the numbers used for counting. They are 1, 2, 3, 4, 5, 6... The symbol... means and so on. That is, the numbers continue on indefinitely according to the pattern of adding 1 to each number to obtain the next number. Whole numbers are the natural numbers and zero. They are 0, 1, 2, 3... Prime numbers are whole numbers other than 0 and 1 that are exactly divisible only by themselves and 1. Examples of prime numbers are 2, 3, 5, 7, and 11. Signed numbers are numbers which have either a positive or a negative sign in front. Numbers preceded by a positive sign are called positive numbers; numbers that are preceded by a negative sign are called negative numbers. Zero is neither positive nor negative. Integers are the positive and negative whole numbers and zero. They are:...-4, -3, -2, -1, 0, +1, +2, +3, +4... 2
A number of signs and symbols are used throughout this module. To summarize them: Sign Meaning + addition or positive number - subtraction or negative number x multiplication division < less than, as -3 < -1 > greater than, as +4 > -5 = equal to, as 7 x 6 = 42 (), {}, [] brackets (brackets side-by-side can also serve to indicate multiplication, as (3)(4) = 12) THE NUMBER SYSTEM All Numbers Have Names Numbers are symbols. They answer the question: how many? They may be applied to any kind of object without discrimination. One may say: I count six, or There are one hundred. The engineer usually applies them to one object, speaking of a turbine rating of 100 MW (megawatts), or calculating the heat units in a certain quantity of water as equal to 1000 J (one thousand joules). In elementary manipulation of numbers, the term digit may be used to designate fingers and a number less then ten. Digits are grouped into word names of numbers. Hyphens are used to find word names for numbers such as 29, 52, 75: 29 is written twenty-nine 75 is written seventy-five The Decimal System A number system has a rigid structure. Power engineers use the decimal structure or system for much of their work. It is a number system scaled in units of ten. It is known as decimal notation because it is based on the Latin word for ten: decem. Ordinary whole numbers are decimal numbers (decimals, for short). 3
For example, the number 4542 is made up of 4000 + 500 + 40 + 2, or four places of digits. The value of any place is ten times that to its right. 4542 = 4 x 1000 + 5 x 100 + 4 x 10 + 2 x 1 = 4 x 10 x 10 x 10 + 5 x 10 x 10 + 4 x 10 + 2 x 1 It is good engineering practice to use a type of shorthand called exponential notation for 10 x 10 x 10 and similar expressions. For 10 x 10 x 10 write 10 3 (read ten cubed or ten to the third power ) For 10 x 10 write 10 2 ( read ten squared or ten to the second power ) 10 1 is equal to 10 and the 1 is rarely shown. 10 0 is equal to 1 (Any number to the power 0 equals 1, which is sometimes called unity ) Thus 3 5 means 3 x 3 x 3 x 3 x 3, 30 4 means 30 x 30 x 30 x 30 If Y stands for any number, Y 4 means Y x Y x Y x Y If Y stands for 2, then Y 8 is equal to 256. The Line Concept of Numbers Numbers may be represented as points on a line: Zero is a number with no value. All numbers to the left may be thought of as negative numbers: -1, -2, -3, -4, -5, etc. All number to the right of the zero are then positive numbers, usually written without the positive sign (+) but the positive sign is always understood to exist. 4
The Place Value of Numbers A chart is a convenient method of recognizing place values of numbers, whether positive or negative, and their names. Table 1 Place Values The example shown below Table 1 is written: 21 505 920 065 and is read: Twenty-one billion, five hundred five million, nine hundred twenty thousand, sixty-five. 716 543 212 is read: Seven hundred sixteen million, five hundred forty-three thousand, two hundred twelve. WHOLE NUMBERS Addition of Whole Numbers Addition is basically a system of counting. The sum 3 + 4 is determined by counting first a set of 3 objects, then another set of 4 objects, joining the two sets, and counting all the objects. In the statement 3 + 4 = 7, 3 is called an addend, 4 is an addend, and 7 is their sum. To add large numbers, place the ones under ones in one column, and tens, hundreds, and so on are forced to fall in their proper columns. Example 1: (a) Add 17 + 5 + 123 5
Solution: 17 The ones are added first: 7 + 5 + 3 = 15, write 5 5, carry 1 to the tens 123 The tens are added: 1 + 0 + 2 = 3 and 1 from 145 (Ans.) the ones column; write 4 The hundreds column: 1 + 0 = 1, write 1 (b) 84 92 127 303 (Ans.) (c) 123 1 526 17 825 19 474 Ans. It is considered desirable to separate larger numbers into groups of three, with a space between groups. A space is not necessary with four digit numbers except when they are in a column and must be aligned with larger numbers. Subtraction of Whole Numbers Subtraction undoes addition. We subtract to get the difference between two numbers, the difference between two sets of numbers. The minus sign (-) is placed between two numbers to indicate that the second number must be subtracted from the first. Larger numbers are arranged as in addition. Ones are subtracted first, then tens, then thousands, and so on. Also, 6778 is the same as: 6000 + 700 + 70 + 8-2314 -(2000 + 300 + 10 + 4) 4464 4000 + 400 + 60 + 4 37 is the same as: 20 + 17-18 -(10 + 8) 19 10 + 9 Example 2: (a) Subtract 18 from 37 6
Solution: 37 Borrow from the tens column, read 17-8, write 9-18 Subtract 1 from the 3 to make up for the 1 19 borrowed from the tens column. Read 2-1, write 1 (b) 1654-965 689 (c) 87 051-32 499 54 552 Multiplication of Whole Numbers In multiplication, two numbers, called factors, are counted to get a third number, called a product. The number which is multiplied is called the multiplicand. The number by which it is multiplied is called the multiplier. The resulting number is called the product. Thus 3 x 5 is found by counting 3 sets of 5 objects each, joining them, and counting them all: 3 x 5 = 5 + 5 + 5 = 15 When a number must be added to itself several times, the process may be shortened considerably by multiplication. Multiplication tables, such as Table 2, are used and must be memorized. MULTIPLICATION TABLE X 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 Table 2 - Multiplication Table 7
Again numbers must be carried to be placed under the correct tens, hundreds, etc., columns as in addition. 827 x 9 = 7443 is arrived at as follows: 9 x 7 = 63, write 3 and carry 6 to the tens column 9 x 2 = 18 plus the 6 carried over, making 24; write 4, carry 2 9 x 8 = 72 plus the 2 carried over, making 74; write 74. Example 3: (a) 34 (multiplicand) x 6 (multiplier) 204 (product) (b) 37 x 29 333 740 1073 (c) 1364 5004 5456 6820000 6825456 The product of any number multiplied by zero is zero. Division of Whole Numbers Division is a sharing. The sign for division is and it is read divided by. The expression 15 5 means that 15 objects are shared or divided into 5 sets. Division is the process of calculating how many times one number is contained in another. It is the converse of multiplication. The number that is divided is the dividend, the number it is divided by is the divisor and the answer is the quotient. 8
Thus dividing 34 916 by 7: 4988 7 ) 34 916 28 7 will not divide into 3, but will go into 69 34. We try the largest number of times it 63 will go, put this in the quotient, write the 61 result under 34 and subtract. Carry the 9 56 down as shown and continue the solution. 56 56 0 It often happens that division cannot be carried out exactly. When the divisor does not go an exact number of times into the dividend, the excess is called the remainder. In power engineering, division by zero is excluded; we never divide by 0. Also, zero divided by any number is zero. DECIMAL NUMBERS Reading and Writing Decimal Numbers A decimal fraction is a fraction which has 10 or a power of 10 as a denominator. Thus 26 or 76 or 532 are decimal fractions. 10 100 10 000 In writing a decimal fraction, it is convenient to omit the denominator and indicate what it is by placing a point or period (called a decimal point) in the numerator so that there shall be as many figures to the right of this point as there are zeros in the denominator. The above fractions now become 2.6, 0.76 and 0.0532, read as two decimal six, zero decimal seven-six, and zero decimal zero-five-three-two. The zero is sometimes omitted. The term common is used to describe all fractions other than those which are decimal fractions. For example 1/2, 3/4, etc., are common fractions, their equivalents 0.5 and 0.75 are decimal fractions. To change 1/2 to 0.5 the rule is: Add zeros to the numerator and divide by the denominator. Place the decimal point so as to make as many digits to the right of the decimal point as there were zeros added. 9
For example, to change, 3/4 to a decimal add two zeros to the numerator and divide by the denominator. 75 4 ) 300 28 20 20 0 Write the result as 0.75. (There are two digits to the right of the decimal.) Notice that when the decimal number has no digits to the left of the decimal point, a zero is placed to the left of that decimal point. Ordering of Decimal Numbers Table 3 shows the order and names of the places to both the left and the right of the decimal point, with the number 765 432.135 79 illustrating the simplicity of the decimal number system. Hun Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Decimal Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths 7 6 5 4 3 2. 1 3 5 7 9 Table 3 Decimal Number System The decimal point is located between the ones and tenths places, where whole numbers end and fractions begin. Thus: 1 3 5 7 9 765432.135 79 = 765 432 + + + + + 10 100 1000 10 000 100 000 = 765 432 + 13 579 100 000 10
Addition and Subtraction of Decimal Numbers Adding or subtracting with decimals is similar to adding or subtracting with whole numbers. The important difference is the decimal point. The simplest way to add or subtract is to first line up the decimal points. You may find it easier to write extra zeros to the right of the decimal point so that the numerals have the same number of decimal places. Carrying figures from column to column in the presence of a decimal point does not affect the procedure. Example 4: Find the sum of 60.95 + 4.0604 + 314.1 + 330 Solution: 60.9500 4.0604 314.1000 330.0000 709.1104 (Ans.) Example 5: Subtract 3.1416 from 8.1 Solution: 8.1000 3.1416 4.9584 (Ans.) Multiplication of Decimal Numbers 1. Location of the Decimal Point To multiply using decimals, multiply as though both factors were whole numbers and then place the decimal point in the result. The sum of the decimal places in the factors is the number of decimal places in the product. 11
For example: 0.035 x 3 = 0.105 0.7 x 7 = 4.9 0.0984 x 0.06 = 0.005904 0.073 x 0.1 = 0.0073 2. Multiplying by 10 and Power of 10 To multiply by 10, merely move the decimal point one place to the right. To multiply by 100 (100 = 10 2 ), move the decimal point two places to the right. To multiply by 10 000 (10 000 = 10 4 ), move the decimal point four places to the right. Division of Decimal Numbers The best method of division by the beginner is to move the decimal point in both the divisor and dividend so that the divisor becomes a whole number. Thus 21.7 = 217 000 = 15 500 0.0014 14 In this case, the decimal point was moved four places to the right. The decimal place must always be moved the same number of places in both the numerator and denominator. Example 6: Find the value of 0.0867 0.21 12 Solution:.4128 0.21)0.0867 = 21)8.6700 84 27 21 60 42 180 168 12
or say 0.413 (Ans.) Notice that the quotient is written over the dividend so that the quotient decimal point comes over the dividend decimal point. As many zeroes as are required can be added to the dividend without affecting its value. Write large numbers in groups of three starting with the decimal point. Four digit numbers that are not placed in a column of larger numbers need not be so spaced. Thus 1 234 567.890 98 is acceptable. If necessary, 0.7854 may be written as 0.785 4 if it is in a column of large numbers. OPERATIONS WITH SIGNED NUMBERS It was stated earlier in the module that the + and - signs have two meanings: + may indicate the operation of addition or it may indicate a positive number; - may indicate the subtraction operation or it may indicate a negative number. Signed numbers can be added, subtracted, multiplied, or divided. A few examples will be looked at first, then the rules pertaining to signed numbers will be summarized. Example 7: (a) (+3) + (+4) = +7 (b) (-4) + (-8) = -12 (c) (+6) + (-2) = +4 (d) (+4) + (-7) + (-5) + (+10) + (-32) + (+9) = (+4) + (+10) + (+9) + (-7) + (-32) + (-5) = (+23) + (-44) = -21 (e) (+2) - (+4) + (-3) - (-6) - (-9) = (+2) + (-4) + (-3) + (+6) + (+9) = (+17) + (-7) = +10 (f) (-5)(-2) = +10 (g) (-18)(5) = -90 13
(h) (+3)(-3)(-4)(-2) = (-9)(-4)(-2) = (-9)(+8) = -72 (i) (-2) 4 =(-2)(-2)(-2)(-2) =(4)(4) = 16 (j) (-1) 113 = -1 (k) (-80) (-20) = 4 (l) (27) (-3) = -9 Summarizing the rules relating to operations with signed numbers: 1. When adding numbers which have the same sign, add the numbers and place the common sign in front of the sum. 2. When adding numbers which have opposite signs, subtract the smaller number from the larger number and assign to the difference, the sign of the larger number. 3. When adding more than two numbers, there are two different techniques which can be used. The numbers can be added two at a time from left to right; or, the positive numbers can be added, then the negative numbers can be added, and finally rule 2 above can be used. 4. When subtracting signed numbers, first change the sign of the number to be subtracted. Then change the subtraction sign to an addition sign. Finally, use the rules of addition to evaluate the expression. (A shorter version of this rule is change the sign of the number to be subtracted and then add. ) 5. When multiplying signed numbers, if the numbers have the same sign, their product is positive. If the numbers have opposite signs, their product is negative. 6. When multiplying several signed numbers, if there is an even number of negative numbers, then the product is positive. If there is an odd number of negative numbers, then the product is negative. 14
7. If a negative number is raised to an even power, then the result is positive. If a negative number is raised to an odd power, then the result is negative. 8. When dividing signed numbers, if the numbers have the same sign, their quotient is positive. If the numbers have opposite signs, their quotient is negative. It should be noted here that zero divided by any number is zero, and any number divided by zero is undefined. that is: 0 = 0-17 and 13 = undefined 0 Order of Operations In the previous section, examples showed situations involving only one basic operation: addition, subtraction, multiplication or division. When evaluating an expression that involves all operations, the order of the operations is most important. The order of carrying out the operations is as follows: 1. First, all powers (exponents) should be evaluated. 2. Secondly, carry out all operations within brackets. If there are brackets within brackets, work from the innermost set of brackets to the outermost set of brackets. 3. Then, carry out any multiplication or division operations working from LEFT to RIGHT. 4. Finally, carry out the addition and subtraction operations. Some examples will help to illustrate the use of these operations in the proper and stated order. 15
Example 8: Simplify (42 + 6) x 9-32 + 16 4-8 Solution: 1. There are no exponents, therefore proceed to point 2 (brackets). 48 x 9-32 + 16 4-8 2. Perform multiplication and division from left to right. 432-32 + 4-8 3. Finally, perform the additions and subtractions. 432-32 + 4-8 = 400 + 4 8 = 404-8 = 396 (Ans.) Example 9: Simplify [-(4 2 ) 2 + (2)(6) - (-6)] (-1)(2) Solution: = (-16 2 + 12 + 6) (-1)(2) = (-8 + 12 + 6) (-1)(2) = (10) (-1)(2) = (-10)(2) = -20 (Ans.) Example 10: Evaluate 27 (-3) + 3{2(-3) 2 - [(-2) 3-3(4)]} Solution: = 27 (-3) + 3{2(9) - [-8-12]} = 27 (-3) + 3 {2(9) - [-20] } = 27 (-3) + 3 { 18 + [+20] } = 27 (-3) + 3 {38} = -9 + 114 = 105 16
Self Test Directions: Answer the following questions. Compare your answers to the enclosed answer key. If you disagree with any of the answers, review learning activities and/or check with your instructor. 1. 32 + 41 + 70 =. 2. 73 + 1204 + 513 =. 3. 82 + 34 + 147 + 1009 =. 4. 76 841 + 39 001 + 967 860 + 4362 =. 5. 1001 + 9009 =. 6. 491-62 =. 7. 1847-792 =. 8. 31 790-21 119 =. 9. 1505-999 =. 10. 999 990-888 889 =. 11. 742 x 397 =. 12. 542 x 238 =. 13. 2345 x 2142 =. 14. 8912 x 106 =. 15. 18 9 =. 16. 75 406 32 =. 17. 112 269 486 =. 18. 5 =. 3-3 17
19. 2 x 4 x 4 =. 12 - (4 x 3) 20. 200 x 3.714 =. 21. 0.012 x 0.04 =. 22. 250 x 0.125 =. 23. 1.75 x 3.5 =. 24. 100 x 0.095 =. 25. 16.0024 8 =. 26. 0.86 2 =. 27. 0.250 0.005 =. 28. 1.728 1.2 =. 29. 1.69 130 =. 30. 12 + 6 3 =. 31. (12 + 6) 3 =. 32. 2 x [36 (18-9 ) + 25-3 2 ] =. 33. 3 x 20 - [13 - (9-3)] =. 34. 40-2 3 4 x 2-5 + 3 =. 18
Self Test Answers 1. 143 18. Impossible 2. 1790 19. Impossible 3. 1272 20. 742.8 4. 1 088 064 21. 0.00048 5. 10 010 22. 31.25 6. 429 23. 6.125 7. 1055 24. 9.5 8. 10 671 25. 2.0003 9. 506 26. 0.43 10. 111 101 27. 50 11. 294 574 28. 1.44 12. 128 996 29. 0.013 13. 5 022 990 30. 14 14. 944 672 31. 6 15. 2 32. 40 16. 2356 with 14 remainder 33. 53 17. 231 with 3 remainder 34. 34 19
References and Reference Material For more information on this topic, the following is recommended: 1. Zimmer, Rudolf A. Essential Mathematics. Dubuque, Iowa: Kendall/ Hunt; 1983. 20