1、,The beginning of our story begins with nothing, absolutely nothing. Well, there was something. Something we know well, but treat it as nothing.,O,Nothing has another name. Its called zero. Zero is something and nothing at the same time. Zero requires nothing. And since there was nothing, zero exist
2、ed. It also existed without needing someone or some thing to make it,O,If zero existed on its own, then a small part of mathematics existed with it. Matter of fact, zero sits right in the middle of mathematics. Its even called the point of origin,O,If zero existed on its own, then a small part of ma
3、thematics existed with it. Matter of fact, zero sits right in the middle of mathematics. Its even called the point of origin,-2 -1 O 1 2,Mathematics has positive and negative numbers with the power to cancel each other out. Start with a positive 1 and combine that with a negative 1, you have zero.,+
4、1,-1,O,Math can give an illusion of something when theres nothing.,You might think the easiest way to expand our mathematical universe is to count: 1, 2, 3, 4, 5 and so on. Counting is actually fairly sophisticated. For example, if you see five items and count them by going, “1, 2, 3, 4, 5“ you have
5、 actually added them up because 5 is the total of all the items.,0, 1, 2 , 3, 4, 5, .,4,5,1,3,2,A more simple expansion would be to say, “1 and 1 and 1 and 1 and 1 . You dont have to know how to count or add.,0 1 1 1 1 1,1,1,1,1,1,Before humans knew how to count, they still were able to keep track o
6、f their possessions. For example, if a sheepherder had 10 sheep, he would pick up one stone for the first sheep, a second stone for the second sheep, a third stone for the third sheep and so on. Hed place on the stones in a leather pouch. At the end of the day he wanted to see if any sheep was missi
7、ng, he didnt count them, he opened the pouch and took out one stone for each sheep he saw. If there were any stones left in the pouch, he knew that not all sheep were present, and would go out to look for them. Another word for stone is calculus. So our advance math called “calculus“ gets its name f
8、rom the simplest of math ideas- one to one correspondence. with rocks.,But Lets get back to our mathematical universe. It expands with simple repetition. Of course, the negative counterpart is also being created one at a time to keep it balanced to zero. We can imagine a string of ones emanating fro
9、m zero,-1 -1 -1 -1 0 1 1 1 1,1,1,1,1,-1,-1,-1,-1,.but to balance the impulse for fast expansion, there ought to be a slow weak attraction force that will pull it all back together. Something like gravity.,When things attract they come together. This action of coming together is called addition. Addi
10、tion doesnt always give us larger numbers. Remember when positive and negative amounts cancel.,1,1,1,1,-1,-1,-1,-1,1,1,-1,-1,The universe to follow will have all kinds of situations where things combine. Adding is an action upon real quantities. For example, three balls collide with five balls. Thei
11、r quantities are added even if you dont know what the total is. The mathematical universe doesnt care that the “answer” is eight. There are eight there whether or not we know that there are. In other words, mathematics does its own math without humans.,Lets look at more examples of addition in actio
12、n. Lets say these fish weigh 1 kilogram each. There are 10 fish on the pan. How much is their combined weight?,You might say 10 kilograms. Nothing in nature understands what 10 kilograms is. But the total weight is known as soon as the last fish hits the pan.,Humans, however, often want a symbol to
13、represent weight. Picking up the pan and saying, “These fish feel heavy.” may not be enough. Nowadays people look at the scale and write down “fish weigh 10kg.” But note that “10kg” is not the weight of the fish.,The weight of the fish can only be felt but not written down. We can write down 10kg bu
14、t it doesnt weight the same as the fish. Now we could use a balance and add rocks until they weigh the same as the fish. If someone asked how heavy were the fish, we could say, “Pick up that bag of rocks and you will know.”,The point here is that we use symbols to represent quantities, but these sym
15、bols are not the same as the quantities. The symbol 10kg might be meaningless to most people, but the actual weight of 10kg falling on their heads would not be meaningless.,kg, kilogram, lb, pound, ton, g, gram, ounce, oz.,When we write this symbol, “5” it is not the number 5. It is a symbol that so
16、me humans recognize as standing for 5 things. Its more properly called a “numeral.” Somewhere in math education, the symbols for mathematics got confused with as being mathematics.,5,1,1,1,1,1,What fraction is this?,Im sorry, this is not a fraction. The proper name is called a fractional numeral.,Wh
17、en we stare too much at symbols we begin to believe that the symbols are real.,A,A,C,A,B,D,When we stare too much at symbols we begin to believe that the symbols are real.,Symbols are even representing emotions,When we stare too much at symbols we begin to believe that the symbols are real.,Its like
18、 thinking your name is somehow more real than you are. A familys name is far less important than the family itself.,e=mc2 is made of symbols for energy, mass and the speed of light, but this is the real e=mc2,Symbols Quiz,How much money is this? $150. Which is worth more , , or ? H2O is water The fo
19、rmula for water is H2O. H is hydrogen H is the symbol for hydrogen You take someones temperature and the thermometer reads, 104oF. That means the person must have a temperature. True/false?,ADDITION,Addition is an easy concept, but there are a few precautions. Theres a clich which says, “You cant co
20、mpare apples to oranges.” Addition has a similar problem. You cant add apples to oranges to find a total of apples or oranges. However, you can still add them.,= ?,+,A + R = A + R,The beginning of our story begins with nothing, absolutely nothing. Well, there was something. Something we know well, b
21、ut treat it as nothing Nothing has another name. Its called zero. Zero is something and nothing at the same time. Zero requires nothing. And since there was nothing, zero existed. It also existed without needing someone or some thing to make it If zero existed on its own, then a small part of mathem
22、atics existed with it. Matter of fact, zero sits right in the middle of mathematics. So in our story zero and that small part of mathematics existed when nothing else could. Math also gives us an illusion of something when theres nothing. For example, give me a million dollars and at the same time g
23、ive me a debt of a million dollars. The million dollars will look like Im rich and have tons of possessions. However, because of the million dollar debt, I actually have nothing. The two cancel Mathematics have positive and negative numbers with the power to cancel each other out. Start with a posit
24、ive 5 and combine that with a negative 5, you have zero. In our mathematical universe two quantities can pop into existence and just as easy pop out of existence. You might think the easiest way to expand our mathematical universe is to count: 1, 2, 3, 4, 5 and so on. Counting is actually fairly sop
25、histicated. For example, if you see five items and count them by going, “1, 2, 3, 4, 5“ you have actually added them up because 5 is the total of all the items. A more simple expansion would be to say, “1 and 1 and 1 and 1 and 1 . You dont have to know how to count.,Before humans knew how to count,
26、they still were able to keep track of their possessions. For example, if a sheepherder had 10 sheep, he would pick up one stone for the first sheep, a second stone for the second sheep, a third stone for the third sheep and so on. Hed place on the stones in a leather pouch. At the end of the day he
27、wanted to see if any sheep was missing, he didnt count them, he opened the pouch and took out one stone for each sheep he saw. If there were any stones left in the pouch, he knew that not all sheep were present, and would go out to look for them. Another word for stone is calculus. So our advance ma
28、th called “calculus“ gets its name from the simplest of math ideas- one to one correspondence. with rocks. But Lets get back to our mathematical universe. It expands with simple repetition. Of course, the negative counterpart is also being created one at a time to keep it balanced to zero. We can im
29、agine a string of ones emanating from zero, but to balance the impulse for fast expansion, there ought to be a slow weak attraction force that will pull it all back together. Something like gravity. When things attract they come together. This action of coming together is called addition. Addition d
30、oesnt always give us larger numbers. Remember when +5 and -5 came together? We got zero. The universe to follow will have all kinds of situations where things combine. Adding is an action upon real quantities. For example, three balls collide with five balls. Their quantities are added even if you d
31、ont know what the total is. The mathematical universe doesnt care that the “answer” is eight. There are eight there whether or not we know that there are. In other words, mathematics does its own math without humans.,Lets look at more examples of addition in action. Lets say these fish weigh 1 kilog
32、ram each. There are 10 fish on the pan. How much is their combined weight? You might say 10 kilograms. But remember the fish dont care, the pan doesnt care, the universe doesnt care that the total weight is 10 kilograms. To nature, the total is done as soon as the last fish hits the pan. Addition wo
33、rks on its own. Humans, however, often want a symbol to represent the weight of the fish. Picking up the pan and saying, “These fish feel heavy.” may not be enough. A person may read the scale and write down “fish weigh 10kg.” But note that 10kg is not the weight of the fish. The weight of the fish
34、can only be felt but not written down. We can write down 10kg but it doesnt weight the same as the fish. Now we could use a balance and add rocks until they weigh the same as the fish. If someone asked how heavy were the fish, we could say, “Pick up that bag of rocks and you will know.” The point he
35、re is that we use symbols to represent quantities, but these symbols are not the same as the quantities. The symbol 10kg might be meaningless to most people, but the actual weight of 10kg falling on their heads would not be meaningless. When we write this symbol, “5” it is not the number 5. It is a
36、symbol that some humans recognize as standing for 5 things. Somewhere in math education, the symbols for mathematics got confused with as being mathematics. Heres a question: What fraction is this? . Im sorry, this is not a fraction. The proper name is called a fractional numeral. The word, numeral
37、means that it is a symbol that represents a number.,4 inches,10 centimeters,+,=?,If you were building a shelf that needed to be 4 inches high to accommodate a VCR and another 10 centimeters high to accommodate a DVD player, how might you add these quantities?,Add just means to combine; it doesnt mea
38、n to convert.,To save time in adding, humans have taken advantage of their ability to see patterns. We have extraordinary ability to see shapes and patterns. If there is no pattern, its hard. For example, in 2 seconds, tell me how many sticks have appeared. When numbers are grouped into patterns, we
39、 recognize them much faster.,This is an old way of counting based on the five fingers of the hand. In groups of 5, its much easier to see the total.,Another way we group quantities in order to recognize the number faster,By grouping amounts in easy to recognize quantities, addition is simplified,7 +
40、 6,5 + 5 + 3,Grouping can be done in any way you want to.,6 + 8 + 7,(-1) (+1) 0,1 3 2,6 8 7,7 5,Shuffle quantities around,79 + 83 + 77 + 14 80 + 80 + 80 + ?,METER,40 cm + 15 mm = ?,In nature addition happens all the time, but so does subtraction.,1,1,1,1,Subtract,Tract = tractor Tract = traction Tra
41、ct = tractor beam Tract = distract Tract = retract Tract = detract,Subtract,Sub = under Submarine Sub = less than subhuman,8 - 3 5,How can you take 8 away from 5?,5 + 0 = 5 0 = (3) + (-3) 5 + 3 + (-3) 8 + (-3) ,Source of confusion,+ (plus) sign + addition sign Whats the difference? + addition sign i
42、ndicates action of combining. + (plus) sign is completely different.,- (negative) sign - subtraction sign Whats the difference? - subtraction sign indicates action of taking away. - (negative) sign is completely different.,+ (plus) sign,The most common meaning is that of direction. - +10o is in the
43、direction of getting hotter,METER,SQUARE,Cubic decimeter,LITER,METER,CUBIC,LITER,cubic decimeter,cubic centimeter,milliliter,In a way you cant add 3 to 4, unless you make both of the ones first. Of course we have memorized 3+4=7. But in doing the problem with real quantities, we must first show 3 as
44、 three single items and four as 4 single items. We can then get 7 single items.,For example, how many inches of rain would there be in your 50x50 foot backyard in 2 hours, if 1 million rain drops of cc each fall every 30 minutes and the dirt soaks up one liter of water per square meter per hour? Wit
45、h some work we can predict this. Nature doesnt try to predict, it lives in the here and now. As quantities arrive, addition and subtraction are happening simultaneously. The answer will reveal itself automatically at the 2 hour mark.,To save time in adding, humans have taken advantage of their abili
46、ty to see patterns. We have extraordinary ability to see shapes and patterns. If there is no pattern, its hard. For example, in 2 seconds, tell me how many sticks have appeared. When numbers are grouped into patterns, we recognize them much faster.,This is an old way of counting based on the five fi
47、ngers of the hand. In groups of 5, its much easier to see the total.,Stop ignoring reality,Some people treat mathematics like a bunch of tools that stay in the toolbox. They may look at the tools, name the tools, handle the tools, but never use them for their intended purpose, which is to use them i
48、n the real world. How many times have you done a problem like + = ? How many times have you measured,4 ft x 3 ft = 12 sq. ft 4 sq. ft x 3= 12 sq. ft,The mathematical universe does not use symbols for quantities, it only uses real quantities. This makes calculations instantaneous. Throw a rock in the
49、 air. The speed of the rock, the force of gravity, the time it takes to fall to the ground are all calculated as it happens. Actually it doesnt calculate anything, it just lets the forces do what they do.,The mathematical universe does not use symbols for quantities, it only uses the real quantities
50、. This makes calculations instantaneous. For example, If we throw a rock into the air how long will it take before it hits the ground? Gravity, wind resistance, and speed are all accounted for instantly and constantly. The rock hits the ground at exactly the time it should take. Nature doesnt try to predict the future.,
