SECTION 1. READING AND WRITING NUMBERS PLACE VALUE
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1 Ten Millions Millions Hundred-thousands Ten-thousands Thousands Hundreds Tens Ones Decimal point Tenths Hundredths Thousandths Ten-thousandths Hundred-thousandths Millionths SECTION 1. READING AND WRITING NUMBERS PLACE VALUE We use a number system based on ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Since there are only ten digits, we need a way to represent numbers larger than 9 using a combination of our digits. The method we use is based on the concept of place value. The value of each digit depends on its position relative to the decimal point. For example, the digit 7 in 7105 does not mean the same as the digit 7 in 973. This same concept of place value is also used to represent fractions in decimal notation. As the diagram below shows, each position has a name that indicates its value. The decimal point in a decimal number separates whole numbers from their fractional parts. Whole numbers are written to the left of the decimal point and the value of a digit increases by a factor of 10 as it moves to the left in a numeral. We could continue the pattern to include hundred millions, billions, etc., but you probably will not encounter such large numbers in your program applications. Commas are inserted every three places to the left of the decimal point to make it easier to read the number. Numbers that are not whole numbers are written to the right of the decimal point and decrease by a factor of 10 as it moves to the right in the numeral. We could again continue the pattern to include ten millionths, hundred millionths, etc. Commas are not used for positions to the right of the decimal point. 4 5, 6 1 3, Sometimes we need to write numbers out in word form, such as when we write a check. In fact, if there is an error in the presentation of the amount of the check, the written amount is always considered the legal figure. Decimal numbers are expressed in word form exactly as they are read. To read a decimal number, the whole numbers are read as their place value indicates followed by the name of the group. The decimal point in the number is replaced by the word and. The fractional parts to the right of the decimal point are read as whole numbers followed by the name of the last place value position. For example, the number listed under the place value table (45,613, ) would be written as: Forty-five million, six hundred thirteen thousand, seventy-two and eight hundred nineteen thousand four hundred thirty-seven millionths.
2 Note: When machinists talk about numbers in the shop they speak in thousandths for everything. For example, what a math person calls one tenth, 0.1 is called one hundred thousandths by a machinist (0.100). A math person would read the number.01 as one hundredth but a machinist would call the same number ten thousandths (0.010). PROBLEM SET 1.1 Write the following numbers in words. 1. 2, , ,004, Write the following numerals. 9. Six hundred fourteen 10. Seventy-four thousand, nine hundred six 11. Five million, two hundred three thousand, forty 12. Two thousand sixty-one and twelve thousandths 13. One thousand eight ten-thousandths 14. Thirty-eight and two hundred fifty-one hundred-thousandths 15. Sixty thousand, one hundred twenty-five and two thousandths 16. Nine hundred and one hundred four ten-thousandths 17. Two thousand one hundred eighty-three ten-thousandths 18. Three hundred nine and seven hundred fifty-two thousandths
3 SECTION 2. ADDITION AND SUBTRACTION OF WHOLE NUMBERS AND DECIMALS Addition is the process of finding the total or sum of two or more numbers. When adding whole numbers or decimals by hand, it is best to write the numbers which are being added in columnar form, being careful to align the place values of all the numbers involved. For example, if we bought three items and the cost of each item was $9, $7.49, and $15.98, we would add the three numbers together to find the total cost. First, we would write the numbers in columnar form aligning the place values Next, we would add the digits in each column. If the total of a column is greater than 9, we write the right-most digit and carry the remaining digits to the top of the next column. The first column on the right yields a total of 17, so we write the 7 down and carry the 1 to the next column (the digits which are carried over are shown in bold print). This process continues for each column to get the final answer. Also note that the decimal point stays in its same column The answer to the problem is $ Note that the answer has the same measurement units as the numbers which we added. This will be true for all addition problems. In order to add, the measurement units must be the same. Subtraction is the process of finding the difference between two numbers. Again, we will write the numbers in columnar form, being careful to align the place values. The number we are subtracting or taking away is written on the bottom, and the number we are subtracting from is written on the top. For example, a propane cylinder weighs 21.3 pounds. The same cylinder, when empty, weighs 4 pounds. To find out how much propane is in the tank, we need to find the difference between the current weight and the empty weight of the cylinder. In other words, we will subtract 4 from First, we would write the numbers in columnar form aligning the place values (Note that we wrote 4 as 4.0) Subtracting 0 from 3 in the first column gives us 3.
4 Since it is not possible to subtract 4 from 1, we will borrow from the next place value to the left, namely the 2 in the tens place value position. The 2 becomes a 1 and 10 is added to the 1, giving us 11 instead of 1. Now 4 can be subtracted from 11, giving us 7 for a difference in the second column. The blank space to the left of the 4 is considered to be a Just like in addition, the decimal point stays in its column. And just like addition, we can only subtract if the measurement units are the same. Since we subtracted 4 pounds from 21.3 pounds, our answer is 17.3 pounds. Example 2.1 Find the difference between 600 and Solution: The problem in this example arises when we need to borrow, but the number we borrow from is We will borrow from the first nonzero digit, in this case the 6 in the hundreds place value position. The zero in the tens position now becomes a 10. Now we can borrow from the 10 in the tens position, change it to a 9, and have 10 in the ones position Next we will borrow from the 10 in the ones position, change it to a 9, and have 10 in the tenths position Finally we can subtract each column. Remember to carry the decimal point down in its column The difference between 600 and is
5 PROBLEM SET 1.2 Find the sums or differences as indicated. In problems where the measurement units are given, make sure you include the proper measurement unit in the answer. 1) ) ) 25 in in in. 4) 5287 gal gal. 5) 7.89 sq. in sq. in. 6) mm mm 7) oz oz. 8) lb lb. Solve the following problems involving addition or subtraction. 9) According to the American Wire Gage table, #8 wire has a diameter of in. and #16 wire has a diameter of in. How much larger is #8 wire than #16 wire?
6 10) What is the wall thickness of a pipe which has an outside diameter of 14 inches and an inside diameter of 12 inches? 11) What is the length of the T-slot bolt shown below? 12) Fifty feet of wire weighs 2.50 pounds and 73 feet of the same type of wire weighs 3.65 pounds. How much does 23 feet of this wire weigh? 13) A combination of gage blocks is selected to provide a total thickness of in. A in block and a in. block are already selected. What is the required thickness of the remaining blocks? 14) A piece of round stock is being turned to a 17.8 mm diameter. A machinist measures the diameter of the current piece to be 18.1 mm. What depth of cut should be made to turn the piece to the required diameter? Hint: The answer is the difference in the radii, not the difference in the diameters.
7 SECTION 3. ROUNDING, ESTIMATING, AND USING A CALCULATOR Exact numbers are not always necessary or desirable. Sometimes it may be necessary to express the number which is a result of a calculation to a place value that is appropriate for the situation. For example, in a problem involving money, the answer is usually expressed to the nearest cent. Therefore, if a calculation produced an answer of $ , the answer would be rounded to $ How a measurement is rounded also conveys the precision of the instrument used to make the measurement. A length measurement using a metric ruler might be 54 mm, indicating that the smallest unit the ruler can measure is 1 mm. A micrometer reading would be given as mm, indicating that this instrument can measure 0.01 mm as its smallest unit. The rounding of a number involves the determination of its significant place value. Any number may be rounded and, quite often, in several ways (i.e., thousands, ones, hundredths, etc.). To round a whole number or decimal number to any place value, we will be using the following steps: ROUNDING A NUMBER STEP 1 STEP 2 STEP 3 STEP 4 Identify the place value (precision) desired in the number to be rounded. Look at the first digit to the right of the selected place value to be rounded. If this digit is 4 or less, leave the selected place value digit the same. If this digit is greater than 5, increase the selected place value digit by 1. If the number is exactly 5 and more digits following the 5, increase the selected place value digit by 1. If the number is exactly 5 with no other digits following the 5, round the selected place value to the nearest even digit. Change all digits to the right of the rounding digit to zeros. Drop all trailing zeros to the right of the decimal point. Example 3.1 Solution: Round to the nearest thousandth. Since the digit 5 is in the thousandths place value position, we look to the digit to the right which is an 8 in this example. Since 8 is greater than 5, we round the 5 up to a 6. We now have following step 3. The last step is to drop the two trailing zeros, giving us a final answer of rounded to the nearest thousandth.
8 Example 3.2 Solution: Round to the nearest hundred. Since the digit 1 is in the hundreds place value position, we look to the digit to the right which is a 2 in this example. Since 2 is less than 5, we do not increase the 1 digit. Performing step 3 gives us Dropping the trailing zero to the right of the decimal point gives us our final answer, which is 100 rounded to the nearest hundred. Example 3.3 Solution: Round to the nearest tenth. Since the digit 9 is in the tenths place value position, we look to the digit to the right which is a 5 in this example. Since we round up when the digit is 5 or more, we round the 9 up to a 10. Since we can t write 10, we write the zero and carry the one to the next place value. We now have following step 3. The last step is to drop the four trailing zeros, giving us a final answer of rounded to the nearest tenth. (Note how we left the zero in the tenth place value position. Without that zero, it would have appeared as though we rounded to the nearest whole number rather than the nearest tenth.) A practical use of rounding is the process of estimating answers to calculations. In order to get a rough idea of an answer, we can round each number to the first nonzero digit in the number and perform the indicated operation. For example, suppose we purchased three items costing $21.55, $6.75, and $ To get an estimate of the total cost, we would round the costs to be $20, $7, and $10. Since is easier to calculate in our head than the original numbers, we have an estimate of $37. The actual answer is $40.33 which is close to our estimate. The rest of the unit will ask us for an estimate and an actual answer to give us practice in finding an estimate before an actual answer is found. We are living in a world where electronic calculators are commonly used in finding answers to calculations. A long division problem that may take several minutes to perform by hand will be calculated in several seconds using a calculator. However, the calculator does have a drawback. It cannot think. Only the operator of the calculator has the ability to think. The abuse of calculator usage happens when the user applies the calculator without having any idea of what kind of an answer they would expect. For example, suppose the calculation was An estimate for the difference involves rounding each number, namely 80 20, which gives us an approximate answer of 60. If we accidentally hit the division key rather than the subtraction key on our calculator, the calculator gives me an answer of A person would have given an obvious wrong answer had he not estimated first and expected an answer close to 60. Some employers give pre-employment tests to prospective employees. Many of these tests contain basic math skills of adding, subtracting, multiplying, and dividing whole numbers, decimals, fractions, and percents without the use of a calculator.
9 So we encourage you to practice enough problems in the course to make sure you do not forget these skills, but we also realize that employers will expect that you know how to correctly use the calculator to increase productivity. And the correct use of the calculator involves estimating answers to avoid writing obvious wrong answers that arise because of data entry mistakes. PROBLEM SET Round to the nearest ten. 2. Round to the nearest thousandth. 3. Round 2,180.7 to the nearest hundred. 4. Round to the nearest thousandth. 5. Round to the nearest thousandth. 6. Round 327,291 to the nearest thousand. 7. Round to the nearest thousandth. 8. Round to the nearest ten-thousandth. 9. Round to the nearest whole number. 10. Round 57,295 to the nearest ten-thousand. 11. Round to the nearest ten-thousandth. 12. Round to the nearest ten-thousandth. 13. Round to the nearest thousandth. 14. Round to the nearest thousandth. 15. Round to the nearest whole number. 16. Round to the nearest ten-thousandth. 17. Round to the nearest hundred-thousandth. 18. Round to the nearest ten.
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