| 1 | From Counting to Addition |
We’ve all seen or heard people count. You put a number of things in a group, move them to another group one at a time, and count as you go. “One, two, three …” We learn to save time when counting by making up groups. Figure 1-1 shows four different ways in which seven items, in this case coins, can be grouped.
Figure 1-1
These are some of the ways seven things can be arranged.
NUMBERS AND NUMERALS
You’ll often hear the terms “number” and “numeral” used interchangeably, as if they mean the same thing. But there’s a big difference (although subtle to the nonmathematician). A number is a quantity or a concept; it’s abstract and intangible. You can think about a number, but you can’t hold one in your hand or even see it directly. A numeral is a symbol or set of symbols that represents a number; it is concrete. In this book, numerals are arrangements of ink on the paper that represent numbers.
COUNTING IN TENS AND DOZENS
When you have a large number of things to count, putting them into separate groups of convenient size makes the job easier. People in most of the world use the quantity we call ten as the base for such counting. How can we show the idea of ten in an unambiguous way? Here’s one method:
• • • • •
• • • • •
There are ten dots in the above pattern. The numbering scheme based on multiples of this number is called the decimal system. The number ten is represented by the numeral 10 in that scheme. Two groups of ten make twenty (20); three groups of ten make thirty (30); four groups of ten make forty (40); and so on, as shown in Fig. 1-2.
Figure 1-2
It’s convenient to count big numbers in tens.
Tens aren’t the only size of base (also called radix) that people have used to create numbering systems. At one time, many things were counted in dozens, which are groups of twelve. This method is called the duodecimal system (Fig. 1-3). We can show the concept of twelve unambiguously, as a set of dots, like this:
• • • • • •
• • • • • •
Figure 1-3
Items can be counted by twelves, but this is not done much nowadays.
WRITING NUMERALS GREATER THAN 10
When we have more than ten items and we want to represent the quantity in the decimal system, we state the number of complete tens with the extras left over. For example, 35 means three tens and five ones left over. The individual numerals or digits are written side by side. The left-hand digit represents the number of tens, and the right-hand digit represents the number of ones. Figure 1-4 shows how this works for 35.
Figure 1-4
Writing numbers bigger than 10. When items are left over, we put the numeral representing them in the ones place.
WHY ZERO IS USED IN COUNTING
If we have an exact count of tens and no ones are left over, we need to show that the number is in “round tens” without any ones. To do this, we write a digit zero (0) in the ones place, farthest to the right. This tells us that we have an exact number of tens because there are no ones left over. “Zero” means “no” or “none.” Figure 1-5 shows how this works for 30.
Figure 1-5
When there aren’t any ones left over, we write zero (0) in the ones place.
BY TENS AND HUNDREDS TO THOUSANDS
One indirect way to visualize numbers is to think of packing things into boxes. A box might hold 10 rows of 10 apples, for example. The total would be 10 × 10, or 100 apples. Then you might stack 10 layers of 100, one on top of the other, getting a cube-shaped box holding 10 × 10 × 10, or a thousand (1000) apples.
If you imagine packing things this way, you’ll find it easy to comprehend large numbers. You might have two full boxes holding 1000 apples each, and then a third box holding five full layers, six full rows on the next layer, and three ones in an incomplete row. This would add up to two thousand (2 × 1000, or 2000) plus five hundred (5 × 100, or 500) plus sixty (6 × 10, or 60) plus three (3), written as 2563. Figure 1-6 shows how this works. You can use the same sort of “boxing-up” concept to envision huge numbers of items.
Figure 1-6
Ten rows of 10 items in each layer is a hundred (100). Ten layers of 100 is a thousand (1000). We can build up large numbers such as 2563 in this way.
DON’T FORGET THE ZEROS
When a count has leftover layers, rows, and parts of rows with this systematic arrangement idea, you will have numbers in each column. But if you have no complete hundreds layers (Fig. 1-7A), there will be a zero in the hundreds place. You might have no ones left over (as at B), no tens (as at C), or no tens and no hundreds (as at D). In each case, it’s important to write a zero to keep the other numerals in their proper places. For this reason, zero is called a placeholder. Never forget to use zeros!
Figure 1-7
Using rows, columns and layers to represent numbers that contain zeros. At A, there are 3065 items; at B, there are 4370 items; at C, there are 2504 items; at D, there are 3008 items.
MILLIONS AND MORE
Imagine stacking thousands of “boxes” to get a powerful way of counting. In Fig. 1-8, one complete box that contains 1000 apples is magnified in a “stack” that has 999 other identical boxes. Think about it: 1000 boxes, each with 1000 apples in it! Each layer of 10 by 10 boxes contains a hundred thousand (100,000) apples. Each row or column of ten boxes in a layer contains ten thousand (10,000) apples. The whole stack contains a thousand thousand or a million (1,000,000) apples.
Figure 1-8
Rows, columns, and layers of 10 items can make a thousand “box.” Rows, columns, and layers of 10 such “boxes” can make a million “stack.”
You can go on with this. Suppose each of the tiniest boxes in the magnifying glass is really a stack containing 1,000,000? Then the 10-by-10-by-10 “superbox” in the glass has a thousand million (1,000,000,000) apples. In the United States this is called a billion. The entire stack in this case has a million million (1,000,000,000,000) apples. In the United States it is called a trillion, but some people in England call it a billion. If you go to another multiple of a thousand, you get a quadrillion (written as a one with fifteen zeros after it). Going onward by multiples of a thousand, you get a quintillion (a one with eighteen zeros after it), a sextillion (a one and then twenty-one zeros), a septillion (a one and then twenty-four zeros), an octillion (a one and then twenty-seven zeros), a nonillion (a one and then thirty zeros), and a decillion (a one and then thirty-three zeros).
WHAT ABOUT INFINITY?
A decillion decillion would be written as a one followed by 66 zeros. A decillion decillion decillion would go down as a one and then 99 zeros. Multiply that huge number by 10, and you get a googol, which is written as a one and then 100 zeros. Of course, we can go on for years with this game, but no matter how long we keep it up, the number we get will be finite. That means that you could count up to it if you had enough time. We can never get an infinite number this way.
No matter how big a number is, you can always get a bigger number by adding one. In that sense, infinity isn’t a number at all. Some people say there isn’t even such a thing as infinity. But enough about that! Now let’s get back to the practical stuff and see how addition relates to counting.
ADDITION IS COUNTING ON
Now that we’ve developed a method of counting, we can start to work on a scheme for calculating. The first step is addition. Suppose you’ve already counted five items in one group and three items in another group. What do you have when you put them together? The easiest way to picture this addition is to count on. Have you seen children doing this using their fingers? They haven’t memorized their addition facts yet!
If you memorize your addition facts, it’s convenient. But nothing is theoretically wrong with counting on. It’s just cumbersome and tedious. You can make an addition table like the multiplication tables you first saw in elementary school. Or you can use a calculator! Calculators add by counting on, but they’re a lot faster than people. Calculators are great for adding large numbers to one another. But for the single-digit numbers, it’s a good idea to memorize all your addition facts. You should know right away, for example, that seven and nine make 16 (7 + 9 = 16).
ADDING THREE OR MORE NUMBERS
Here’s a principle that the people who invented the “new math” gave a fancy name: the commutative law for addition. Put simply, it says that you can add two or more numbers in any order you want. Suppose you want to add three, five, and seven. No matter how you do it, you always get 15 as the answer. Figure 1-9 shows two examples with coins. The commutative law applies to as few as two addends (numbers to be added), up to as many as you want.
Figure 1-9
When adding three numbers, it doesn’t matter which two are added first. At A, we add three and five, and then add seven more. At B, we add seven and five, and then add three more. Either way we get fifteen.
ADDING LARGER NUMBERS
So far, we have added numbers with only a single figure in the ones place. Bigger numbers can be added in the same way, but be careful to add only ones to ones; tens to tens, hundreds to hundreds, and so on. Just as 1 + 1 = 2, so 10 + 10 = 20, 100 + 100 = 200, and so on. We can use the counting-on method or the addition table for any group of numbers, as long as all the digits in the group belong. That is, they all have to be the same place: one, tens, hundreds, or whatever.
So, let’s add 125 and 324. Take the ones first: 5 + 4 = 9. Next the tens: 2 + 2 = 4. Last the hundreds: 1 + 3 = 4. Our result is four hundreds, four tens, and nine ones, which total 449. This process is shown in Fig. 1-10. Notice that we are taking shortcuts. We no longer count tens and hundreds one at a time, but in their own group, tens or hundreds. If you added all those as ones, you’d get sick and tired of it a long time before you were done. You might get careless, and you would have 449 chances of skipping one, or of counting one twice. The shortcuts not only make the process go by quicker, but they reduce the risk of making a mistake.
Figure 1-10
Big numbers are added in the same way as single-digit numbers. Here, we see that 125 added to 324 equals 449.
CARRYING
In the preceding example, we deliberately chose numbers in each place that did not add up to over 10, to make it easy. If any number group or place adds to over 10, we must “carry” it to the next higher group or place. Three examples follow; it will help if you write the digits down in the columns for powers of 10 (that is, the ones place, the tens place, the hundreds place, and so on) as you read these descriptions.
Suppose we want to add 27 + 35. We take the ones first: 7 + 5 = 12. The numeral 1 belongs in the tens place. Now, instead of just 2 and 3 to add in the tens place, you have an extra 1 that appears in the tens place from adding 7 and 5. This extra 1 is said to be carried from the ones place. This carrying process goes on any time the total at a certain place goes over 10. The final result of this addition process, called the sum, is therefore 27 + 35 = 62, because we have 1 + 2 + 3 = 6 in the tens place.
Now what if we want to add 7,358 and 2,763? Start with the ones: 8 + 3 = 11, so we write 1 in the ones place and carry 1 to the tens place. Now the tens: 5 + 6 = 11, and the 1 carried from the ones makes 12. We write 2 in the tens place and carry 1 to the hundreds place. Now the hundreds: 7 + 3 = 10 and 1 carried from the tens makes 11 hundreds. We write 1 in the hundreds place and carry 1 to the thousands place. Now the thousands: 7 + 2 = 9 and 1 carried from the hundreds make 10 thousands. The final sum is therefore 10,121.
Another example: suppose we have to add 7,196 and 15,273. We start with the ones: 6 + 3 = 9. We write 9 in the ones place and nothing is left to carry to the tens. Next, 9 + 7 = 16. We write the 6 and carry the 1 to the hundreds. Now the hundreds: 1 + 2 = 3, and the 1 carried makes 4. We have nothing to carry to the thousands. So in the thousands: 7 + 5 = 12. Now we carry 1 to the ten-thousands place, where only one number already has 1. We add 1 + 1 to get 2 in the ten-thousands place. The final sum in this case is 22,469.
CHECKING ANSWERS
Before calculators made things easy, bookkeepers would use two methods to add long columns of numbers. First, they would add starting at the top and working down. Then they would add the same numbers starting at the bottom and working up. They kept right on doing this once they got calculators because it was a good way for them to check their work! Adding long lists of numbers can get tedious, and it’s easy to make a mistake. You don’t want to do that when you’re dealing with important things such as tax returns.
You see the advantage of using more than one method. The partial sums that you move through on the way are different when you go upward, as compared to when you work down. But you should always reach the same answer at the end. It’s not likely that you would enter the same wrong number twice under these conditions. If you get different answers, work each one again until you find where you made your mistake. (And be happy you found it before the tax man did!)
WEIGHTS
Now let’s get away from pure numbers look at some real-world examples of addition and measurement. How about weight?
Modern scales read weight in digital format. They “spit out the numbers” at you. But scales weren’t always so simple. You might have seen another kind of scale that uses a sliding weight. You have to balance it and read numbers from a calibrated scale. The old-fashioned grocer’s balance worked in an even more primitive way. This device is now considered an antique, but knowing how it worked can help you understand addition in a practical sense.
The simplest grocer’s balance, like the thing you’ve seen the blindfolded “Lady of Justice” holding, had two pans supported from a beam pivoted across a point or fulcrum. The pans were equally far away from the fulcrum on opposite sides. When the weights in both pans were equal, the scales balanced and the pans were level with each other. When the weights were unequal, the pan with the heavier weight dropped and the other one rose. To use such scales, the grocer needed a set of standard weights, such as those shown in Fig. 1-11.
Figure 1-11
Standard weights were combined for use on an old-fashioned grocer’s balance.
Standard or avoirdupois weight, still used in some English-speaking countries, does not follow the more sensible power-of-10 metric system. Instead, it defines 16 drams to an ounce, 16 ounces to a pound, 28 pounds to a quarter, four quarters (112 pounds) to a hundredweight, and 20 hundredweights (or 2240 pounds) to the long ton. (A short ton of 2000 pounds is used by most laypeople in “nonmetric” countries).
USING THE GROCER’S BALANCE
A set of weights for use with a grocer’s balance consisted of those shown at the top of Fig. 1-11. It was only necessary to have 12 of them, unless the grocer wanted to measure more than 15 pounds. With these weights, if the scale was sensitive enough, the grocer could weigh anything to the nearest dram.
Suppose we want to weigh a parcel using a grocer’s balance. Figure 1-12 shows how this process might go. First, we put the parcel in the pan on the left. Then we put standard weights on the other pan until the scale tips the other way. If a 1-pound weight doesn’t tip it, we try a 2-pound weight. Suppose it still doesn’t tip! But then the 2-pound and 1-pound weights together, making 3 pounds, do tip it. So we know that the parcel weighs more than 2 pounds, but less than 3 pounds.
Figure 1-12
An example of how we can weigh a parcel using a grocer’s balance.
Now we leave the 2-pound weight in the pan and start using the ounce weights. Suppose 8 ounces don’t tip the scale. If 4 ounces are added to make 12 ounces, it still doesn’t tip. But when we add a 2-ounce weight, which brings the weight up to 2 pounds 14 ounces, it tips. If the 1-ounce weight is used instead of the 2-ounce weight, the scale doesn’t tip. Now we know that the parcel weighs more than 2 pounds 13 ounces, and less than 2 pounds 14 ounces. If we want to be more accurate, we can follow this method until it balances with 2 pounds, 13 ounces, and 3 drams.
QUESTIONS AND PROBLEMS
This is an open-book quiz. You may refer to the text in this chapter when figuring out the answers. Take your time! The correct answers are in the back of the book.
1. Does it make any difference in the final answer whether you count objects (a) one by one, (b) in groups of ten, or (c) in groups of twelve?
2. Why do we count larger numbers in hundreds, tens, and ones, instead of one at a time?
3. Why should we bother to write down zeros in numerals?
4. What are (a) 10 tens and (b) 12 twelves?
5. What are (a) 10 hundreds, (b) 10 thousands, and (c) 1000 thousands?
6. By counting on, add the following groups of numbers. Then check your results by adding the same numbers in reverse order. Finally, use your calculator.
(a) 3 + 6 + 9
(b) 4 + 5 + 7
(c) 2 + 7 + 3
(d) 6 + 4 + 8
(e) 1 + 3 + 2
(f) 4 + 2 + 2
(g) 5 + 8 + 8
(h) 9 + 8 + 7
(i) 7 + 1 + 8
7. Add together the following groups of numbers. In each case, use a manual method (without using a calculator) first, and then verify your answer with a calculator.
(a) 35,759 + 23,574 + 29,123 + 14,285 + 28,171
(b) 235 + 5,742 + 4 + 85,714 + 71,428
(c) 10,950 + 423 + 6,129 + 1 + 2
(d) 12,567 + 35,742 + 150 + 90,909 + 18,181
(e) 1,000 + 74 + 359 + 9,091 + 81,818
8. How does adding money differ from adding pure numbers?
9. Add together the following weights: 1 pound, 6 ounces, and 14 drams; 2 pounds, 13 ounces, and 11 drams; 5 pounds, 11 ounces, and 7 drams. Check your result by adding them in at least three ways.
10. What weights would you use to weigh out each of the quantities in question 9, using the system of weights for a grocer’s balance? Check your answers by adding up the weights you name for each object weighed.
11. In weighing a parcel, suppose the 4-pound weight tips the pan down, but the 2- and 1-pound weights do not. What would you do next to find the weight of the parcel (a) if you wanted it to the nearest dram; (b) if you had to pay postage on the number of ounces or fractions of an ounce?
12. The yard is a unit of length commonly used in the United States. It has 3 feet, and each foot has 12 inches. How many inches are in 2 yards?
13. In the United States, common liquid measures are the pint, the quart, and the gallon. There are 2 pints to a quart and 4 quarts to a gallon (Fig. 1-13). Suppose that a fleet of cars need oil changes. Three cars require 5 quarts each, two cars require 6 quarts each, and four cars require 1 gallon each. How many gallons of oil does the owner need?
Figure 1-13
Non-metric standard units of liquid measure commonly used in the United States. Illustration for problems 13 and 14.
14. If the owner of the cars in the previous problem can buy quarts of oil at 90 cents and gallons at $3.50, how should he buy the oil to be most economical?
15. Suppose a woman buys three dresses at $12.98 each, spends $3.57 on train fare to get to town and back, and spends $5.00 on a meal while she is there. How much did she spend altogether?
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