Wikijunior:Introduction to Mathematics
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Introduction
This book is designed to teach you, in less than a year, all of the mathematics you need to go to college.
Today, I was eating brunch in a restaurant, and I overheard the mother at the next table talking to her young son, a child of about three. She was coaching him in his numbers and alphabet. It was heartwarming.
But, not everyone has the good fortune to be taught their numbers and letters before entering kindergarten. Children who lack this warm connection between learning and their parents may be handicapped forever. Yet, even handicapped students can accomplish wonders. This book begins at the beginning. It assums you know how to read, nothing else. (If you have trouble reading this, you need to learn reading first, but you can work on math, too, if you have someone to read to you.)
The book is organized in a series of short lessons. Each lesson starts by saying what it is designed to teach. If you already know what that lesson is designed to teach, skip to the next lesson. DO NOT SKIP ANY LESSON YOU ARE NOT ABSOLUTELY CERTAIN YOU KNOW. Every lesson builds on what has gone before.
[edit] Chapter One: Counting
[edit] Counting from one to ten.
You learn to count from one to ten by repetition. You also learn to recognize the relationship between numerals and number words. You learn to count backwards from ten to one. You learn to identify from one to ten objects. One to five objects you should identify on sight (without counting). You should be able to do the same with six to ten objects that are grouped in certain patterns.
Most people learn this skill when very young, but just as their are adults, some very successful, who cannot read or write, there are adults who have never mastered this number skill. It is essential to further progress in mathematics.
If an adult does not know how to read at an adult level, I advise them to read aloud to children. If you are an adult, and are unsure about numbers, teach numbers to a child.
The first skill in this section is to be able to say aloud, as you might say the lyrics of a song you heard often:
one two three four five six seven eight nine ten
Practice this, hundreds of times if necessary, until it haunts you, until if someone says "six" you automatically think "seven eight nine ten".
Practice writing the numerals, and saying their names aloud. The first ten numerals are
1 2 3 4 5 6 7 8 9 10
If you are not already perfect at this, write these ten numerals on ten separate sheets of paper, and practice pulling one out at random and saying the name of the number. If you are teaching this skill to a child, make a set of flash cards with the word on one side and the numeral on the other.
The next skill in this lesson is to count down from ten. As with counting, just repeat until you know by heart:
ten nine eight seven six five four three two one.
When you have done this, if you are thinking in terms of "counting down" and someone says "seven" you should automatically think "six five four three two one".
The final skill in this lesson is recognition of "how many objects" there are in a group of one to five objects. Practice this until it becomes automatic. As soon as you see a group of one to five objects, you should know, without thinking about it, how many objects are in the group. If this does not come easy for you, practice with small groups of objects at home until you can do it without counting or even thinking. Then move on to groups of six to ten objects. You do not need to automatically recognize how many objects are in these groups unless the groups are arranged in one of the following patterns, but you should recognize, on sight, each of the following patterns:
@@@ @@@ six objects
@@@ @@@@ seven objects
@@@@ @@@@ eight objects
@@@ @@@ @@@ nine objects
@@@@@ @@@@@ ten objects
Once you have learnt the numbers one to ten, you can practice identifying a bigger number out of two given numbers. For example:
- (2,3) - 3 is bigger
- (6,4) - 6 is bigger
[edit] Counting from eleven to twenty.
In some languages, Chinese for example, the number words from eleven to twenty follow the same pattern as the number words from twenty to thirty, but this is not the case in English, and so you need to memorize the number words from eleven to twenty. Repeat the following until you know it by heart:
eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen twenty
You do not need to learn this list backwards.
The numerals eleven to twenty are
11 12 13 14 15 16 17 18 19 20
You need to associate the numeral with the word so that the sight of either brings to mind the other. Use flash cards.
You also need to think of 11 as ten plus one, 12 as ten plus two, and so on, so that in mastering these number words, you also begin to master addition facts.
[edit] Counting to twenty by twos.
The next skill is to count "by twos" from two to twenty. When you count by twos, each number is two more than the previous number. This kind of counting is sometimes called "skip counting" Memorize:
2 4 6 8 10 12 14 16 18 20
Say it over and over until you know it by heart.
Note that the word "twos" is plural, not possessive, and does not take an apostrophe.
[edit] Counting to one hundred by tens .
The next skill is counting by tens from ten to one hundred. Each number is ten more than the previous number. Memorize:
ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, one hundred.
Note in particularly, the spelling of "forty". That spelling does not make any sense, considering the way we spell "fourth" and "fourteen", but we must follow the dictionary when it comes to spelling.
Learn to match the names and the numerals until it becomes second nature:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Notice that while the spelling (in the case of forty) does not follow a pattern, the numerals do follow a pattern. That is, if you knew ten, twenty, and thirty, that would not help you guess "forty". But if you knew 10, 20, and 30, you could easily guess 40. This is the wonderful thing about mathematics. It follows patterns, so that after you learn just a few things, you can then guess correctly many more things. This is why you can learn, in less than a year, what it takes twelve or more years to learn in school. School usually uses too much memorization. In this book, you will memorize only what must be memorized, and learn the rest by seeing the pattern.
You need to have an idea of how many one hundred is. One hundred is ten tens. Gather one hundred objects on a table, in ten groups of ten. (Really do this. Don't just think about doing it, do it.) One hundred is big, but not too big. I would never ask you to gather a million objects. It would take days, and they wouldn't fit on the table.
[edit] Counting to one hundred by twenties.
Memorize:
20 40 60 80 100
The main reason this is important is that the twenty dollar bill is in common use in the United States.
Remember, memorize means memorize. Yes, you could figure out how to count by twenties. That isn't good enough. This basic stuff you need to be able to do without thinking, while you are thinking about something more important. Be sure you can count by twenties while you are washing dishes or talking on the telephone. Be sure you could count out the values of five twenty dollar bills while carrying on a conversation with the bank teller.
[edit] Counting to one hundred by ones and by fives.
After twenty, all the whole numbers follow a pattern. After you know that 87 is read "eighty seven" and means eighty plus seven, you know almost all the number words you will ever need to know. I haven't counted them, but I would guess that in all the rest of this book you will need to learn less than a dozen more number words. In other words, you've already learned the most important part of mathematics! Now, if you've never done it before, count out loud to 100. You don't need me to tell you how. You can figure it out for yourself. Now count by fives to 100. I'm not going to tell you how to do it. Find the pattern.
[edit] Who's on first?
When a number word is used as an adjective instead of as a noun, it changes. If the number word is one, two, or three, or if the number word ends in one, two, or three, as does two hundred fifty-three, then "one" becomes "first", "two" becomes "second", "three" becomes "third". When the numbers are given using numerals instead of words, we use superscripts: 1st, 2nd, 3rd, pronounced "first", "second", "third". All other number words and numerals add "th" when used as an adjective, thus, "fourth", "fifth", "sixth", "seventh", "eighth", "nineth", "tenth", "eleventh", and so on. Note that "fourth", the adjective form of "four", is not the same word as "forth", meaning "forward". Also note the odd spelling and pronounciation of "fifth" (the second "f" is silent), and the fact that "eighth" does not have a double "t".
When a number word is used as an adjective, it is sometimes called an "ordinal" number, when used as a noun, a "cardinal" number. Thus 3rd is the third ordianal, while 3 is the third cardinal.
[edit] Zero
The number 0, read "zero", stands for "none at all". In the room where you are reading this, there are zero pink elephants. Note that the numeral for 0 is slightly narrower than the letter capital O. Sometime, people draw a slash through their zeroes, so as not to mix them up with O's. (They can also sometimes draw a slash through capital Z so as not to mix it up with 2.)
[edit] Digits
In our number system, which originated in India and has now spread everywhere in the world because it is so useful, every whole number, no matter how large, can be written with just ten digits. The digits are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
In other countries, people sometimes write the digits differently, but they still use ten digits.
[edit] Thousand, million, billion, trillion.
A thousand is ten hundreds. The number 1572 is "one thousand five hundred seventy-two". Notice that numbers between twenty-one and ninety-nine, except for every tenth number (thirty, forty, fifty, and so on) are written with a hyphen.
Starting with ten thousand, numerals are written with commas, so "ten thousand fifty-six" is written 10,056. Notice that the 0 in 056 means that there are no hundreds in this numeral. In this case, the 0 is called a "place holder". Every three digits in a large number are set off by a comma, starting from the right. If a numeral has one comma, it is read "thousand", if a number has two commas the first comma is read "million", if three commas, the first is read "billion", at least in the United States, while in the United States, if a number has five commas, the first is read "trillion". Other countries have different number names. In practice, very large numbers are often read just by reading their digits. Thus we might read 10,056 as "one zero zero five six".
A million is a thousand thousands. It is a very big number, but a billion is much bigger. The word "billion" is used in two different ways. In America, "billion" means a thousand million. In most of the rest of the world, "billion" means a million million. When communicating large numbers across national borders, it is better to use numerals than words.
In America, a "trillion" means a thousand American billions. In most of the rest of the world, a "trillion" is not used at all.
There are words for even larger numbers, but they are seldom used, so we won't bother with them. For large numbers, just read off the digits. One interesting word for a very large number is the word "google". Look it up if you want to know what number it stands for.
It is hard to grasp the size of very large numbers. Here is one way to put them in perspective. As I write this, the national debt of the United States is about nine trillion dollars ($9,000,000,000,000), while the population of the United States is about three hundred million (300,000,000). If the United States spends a million dollars, then each person's share is less than one cent. If the United States spends a billion dollars, then each person's share is between $30 and $40. If the United States spends a trillion dollars, then each person's share is more than $30,000. As you can see, there is quite a difference between a million, a billion, and a trillion.
[edit] The natural numbers
The whole numbers, starting with 1, 2, 3, ..., are called the "natural numbers". The dot dot dot is called an "elipsis" and means "and so on". By the way, an elipsis always has three dots, no more, no less. The natural numbers go on without end. The word we use for "without end" is "infinity". There are infinitely many natural numbers. We could never write them all.
People do not agree on whether to include 0 in the natural numbers. Older authors often do include 0, thus: 0, 1, 2, ..., while more modern authors generally put zero in a class by itself, and start the natural numbers with 1.
[edit] Chapter two: Addition and subtraction, positive and negative.
[edit] What addition does.
Addition is an operation which allows us to find out how many we have when we combine two or more sets of objects. We can only add like objects, and when we use addition to combine two sets of different types of objects, we need to give them a common name. For example, if we combine three apples with two oranges we have five pieces of fruit (fruit being a common name for apples and oranges). If we add three inches and two feet, we first change the two feet to 24 inches, and then extend the 24 inches by three inches to get 27 inches.
The numbers representing the parts in an addition problem are called terms. (There are older words for these parts, but they are seldom if ever used today.) The answer is called the sum. Thus, in
- 2 + 3 = 5
the terms are 2 and 3 and the sum is 5.
We use addition when we make a deposit to our bank account, when we find how many comic books we now have when we put ten more comic books into our collection, and when we add the price of a coke and a sandwitch plus the tax to see if we can afford to buy lunch.
[edit] Learning to add.
We now come to one of the two hard tasks in elementary mathematics, memorizing addition facts. The other hard task is memorizing multiplication facts. In both cases, there are 100 facts to memorize, though various short cuts can reduce this by half.
Most of mathematics we can figure out, which reduces the need to memorize. We could figure out the addition and multiplication facts, but it would take too long. Once we have them memorized, we can do quick mental arithmetic while driving a car or standing in the check out line in a store.
In most schools, addition facts are taught while children are still young enough to memorize easily, and so few students fail to remember their addition facts. On the other hand, multiplication facts are taught after students are too old to memorize easily, and so many students have trouble memorizing their multiplication facts. If you are a parent, using this book to teach your children, teach them their addition facts when they are five and their addition facts when they are six. (Do not teach both at the same time -- children will get them confused.) Always use exactly the same words in teaching rote additon and multiplication. "Two plus three is five." "Two threes are six."
[edit] Adding one.
In Chapter One, you learned to count to 100. To add one, just count up. 3 + 1 = 4. 10 + 1 = 11.
Since 3 + 1 is the same as 1 + 3, you now know 19 out of the 100 addition facts. By the way, the rule that 1 + 3 = 3 + 1 is an example of the commutation law. Picture the 1 commuting to where the 3 lives, and the 3 commuting to where the 1 lives.
[edit] Adding two.
When you first begin to add, it is tempting to add two by counting up twice. This works, but is too slow. Instead of thinking: to add 3 + 2 I start with three and count four five, you need to memorize "three plus two is five". After you do that, every time you see 3 + 2, the number 5 will instantly pop into your mind. We therefore need to bite the bullet and begin to memorize. You memorize these addition facts by saying them over and over and over until you know them by heart.
"two plus two is four, three plus two is five, four plus two is six, five plus two is seven, six plus two is eight, seven plus two is nine, eight plus two is ten"
Read this twenty times a day until you've got it. Flash cards are another good way to learn addition facts. Keep practicing, and at some point the addition facts will move from short term memory (where we keep locker combinations) to long term memory (where we keep the names of our parents). Then you will have them forever, and the effort will be worth while.
Teachers in every grade should make sure that all of the their students know all of their addition and multiplication facts.
You didn't need to relearn one plust two is three. And the commutative law almost doubles the new facts you have learned, so you have learned 13 new fact which together with the 19 facts about adding one means you have learned 32 of the 100 addition facts.
[edit] Adding a number to itself.
For the sake of variety, let's next learn how to add a number to itself. You already know that one plus one is two and that two plus two is four. Here are the rest of those facts. Memorize them.
"three plus three is six, four plus four is eight, five plus five is ten, six plus six is twelve, seven plus seven is fourteen, eight plus eight is sixteen, nine plus nine is eighteen."
You now know 39 of the 100 addition facts.
