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Text of the page (random words):
tion see that the a c and the b d can be switched with no effect on the order of the two terms if that is not obvious the following rearrangement is clear indeed i calculated the two arrangements of the associative rule and it was also satisfied do this yourselves leave a comment filed under uncategorized january 7 2019 1 15 am subtraction using addition subtraction with adding 4531 3726 add 3 to both items 6 and 3 makes 9 4534 3729 add 70 to both items 70 and 20 makes 90 4604 3799 add 200 to both items 700 and 200 makes 900 4804 3999 result 804 1 805 2 comments filed under uncategorized september 2 2018 1 19 am subtraction and negative numbers part_2 using the and connective in apple words we have start with 7 apples subtract 3 apples result 4 apples or as a single sentence starting with 7 apples and subtracting 3 apples gives the result 4 apples or without the apples start with 7 and subtract 3 gives the result 4 using the minus sign in place of subtract we can simply replace subtract by minus in the subtract 3 apples expression as they have the same meaning and we can use the minus sign instead of the word so we have converted subtract into the minus sign and correspondingly add into the plus sign and used the word and to link the terms together beginning with the example on the previous post start with 9 add 2 subtract 3 subtract 5 add 4 subtract 2 final state 5 we get the expression on the left start with 9 and 2 and 3 and 5 and 4 and 2 and since the and connectives are all of the connectives we can dispense with them and write the expression as start with 9 2 3 5 4 2 with result or final state 5 we can even cross out the start with at the beginning as it is always there and simplify using the arithmetical equals or 9 2 3 5 4 2 5 multiplication and division with numbers an example start with 5 add 6 multiply by 4 multiply by 2 subtract 21 result is 32 5 a nd 6 multby 4 multby 2 and 21 5 6 4 2 21 result 32 at this point there appears to be a problem do we multiply out the factors multbys before the sums are calculated or afterwards the convention is to multiply the factors in each group of multbys first and then combine and the add and subtract items for the above calculation the result develops as follows 5 6 4 2 21 32 5 6 8 21 32 5 48 21 32 32 and with two groups of mults 5 6 3 1 2 4 12 28 5 18 1 2 4 12 28 5 18 1 8 12 28 28 and now we have to do algebra algebra expressions terms and factors a term in algebra is not just a number preceded by a or a it is a number which may be 1 and can be ignored followed if needed by a sequence of letters variables and expressions each of the expressions is enclosed in brackets parentheses an algebraic expression is startwith 0 followed by a list of terms simple example startwith 0 8a bx the startwith number is always zero and we can ignore it leaving only the terms the first term has a or a and we can ignore the first as 8a bx if the terms are rearranged then the missing needs to be put back in as in bx 8a examples 8a bx 4 3x 7xy 4 3x 7xy and with included expressions like x 5 4 3x x 5 2 5x 6 x 2 x 9 the value of an expression is the numerical result of the expression when all the individual variables are replaced by numbers and combined an equation is two expressions with an equals sign between them if the value of the left side of an equation is equal to the value of the right side then the equation is satisfied the number on the left equals the number on the right examples x 3 4x 6 the equation is satisfied when x 3 and nowhere else 3x a 2x where x 6 now find a to satisfy the equation the distributive property of multiplication over addition this is the most important feature of arithmetic and algebra where the additive form of an expression is converted into a multiplicative form and vice versa four examples will show the methods a x y ax ay ax bx a b x a b x y ax ay bx by a b x y ax ay bx by this last is a bit obscure but a two dimensional array will help more tricky details for exponents and for division in algebra exponents simple examples 2 3x 2 3 x 5 4x 6 x 2 the two examples above are the conventional forms of expressions with exponents the two below are fully bracketed expressions showing the linked pairs of parentheses and showing the signs and numbers for the exponents is the sign for exponent 2 3 x 2 3 x 5 4 x 6 x 2 these are both one line expressions division simple examples the two examples above are the conventional forms of expressions with division the two below are fully bracketed expressions showing the linked pairs of parentheses and showing the signs and numbers for division is the sign for divide by 6 7 x 6 7 y x 2 these are also both one line expressions it will appear that with the apple construction the formulae for arithmetic and algebra are identical leave a comment filed under uncategorized may 25 2018 11 13 pm subtraction what exactly is it part 1 bonus code for minus is 8722 addition i had 5 apples and then my brother gave m 2 more apples i now have 7 apples the process is without the pictures starting state 5 apples action add 2 apples and bunch them up final state 7 apples with a single sentence or equation with the result start with 5 add 2 and resulting state is 7 or the equation with apples start with 5 apples add 2 more apples and result is 7 apples subtraction i have 7 apples now and then my brother takes 3 of them away i end up with 4 apples the process is without the pictures starting state 7 apples action subtract 3 apples final state 4 apples with a single sentence or equation and the result start with 7 subtract 3 and resulting state is 4 or the equation with apples start with 7 apples subtract 3 apples and result is 4 apples both of these equations are read from left to right counting and adding if i start with 1 and then add 1 i get 2 this doesn t seem to be a problem at all if i start with 2 and then add 1 i get 3 this doesn t seem to be too big a problem if i start with 5 and then add 3 i get 8 this starts to get interesting because starting with 2 and adding 5 has exactly the same final state it is all to do with counting if i count a pile of bricks and there are ten of them i can jumble the bricks around and count them again i will still have ten of them the count is how many and nothing else consequently the rearrangement has no counting effect and so any collection of numbers with plus signs and a total of ten can be rearranged to give the same total of ten for example start 2 add 3 add 4 add 1 is the same total as start 2 add 4 add 1 add 3 result ten we can also swap the start number and the add number with an example start 3 add 5 is start 4 add 4 and then start 5 add 3 same total and finally with add we can combine two adds to get add 3 add 2 add 5 counting and subtracting the apple example starting state 7 apples action subtract 3 apples final state 4 apples the starting state or start number is 7 the action is subtract 3 and the final state is 4 subtracting several numbers have a look at the following 9 and 2 and 4 the final state is 3 but so is 9 and 4 and 2 consequently the two subtractions can be swapped it s even better than this the adds and the subtracts can be put in any order with the same final state try start with 9 add 2 subtract 3 subtract 5 add 4 subtract 2 final state 5 or rearranged start with 9 add 2 add 4 subtract 3 subtract 5 subtract 2 final state 5 properties of operations a critique operations and algebraic thinking ccss grade 1 italics represent and solve problems involving addition and subtraction 1 2 understand and apply properties of operations and the relationship between addition and subtraction 3 apply properties of operations as strategies to add and subtract examples if 8 3 11 is known then 3 8 11 is also known commutative property of addition to add 2 6 4 the second two numbers can be added to make a ten so 2 6 4 2 10 12 associative property of addition in mathematics there are numerous algebraic structures each consisting of a set of elements equipped with an operation that combines any two elements to form a third element wikipedia adapted 19th century math so we can find an example of this where any two elements in the set of natural numbers can be combined with with the usual interpretation 7 4 11 more specifically we can use the commutative property of addition a feature of the natural number system to tell us that 7 4 equals 4 7 for example but take a look at the above example with different numbers 7 6 4 swap 4 and 6 7 4 6 so we have to avoid the negative operator the conclusion is that the commutative rule applies only when all the operators are plus operators now what are we going to make of this table 3 the properties of operations here a b and c stand for arbitrary numbers in a given number system the properties of operations apply to the rational number system the real number system and the complex number system ccss commutative property of addition is a b b a this is a definition it states that a complicated name is what we know already and can actually prove by counting so it is a complete waste of time for k 7 or further not only that but the actions add 3 and subtract 4 are the original 17th and 18th century statements and are proper algebra leave a comment filed under uncategorized image wiggly line leave a comment april 6 2018 10 21 pm february 17 2018 8 25 pm the chain rule and the theory i found this account of the chain rule from sam shah https samjshah com 2018 02 08 there might be light at the end of the chain rule tunnel maybe the partial text is as follows find two points close to each other like x g f x and x 0 001 g f x 0 001 find the slope between those two points g f x 0 001 g f x x 0 001 x there we go an approximation for the derivative we can use limits to write the exact expression for the derivative if we want but that doesn t help us understand that d dx g f x g f x f x on any level they seem disconnected but i m on my way there i m following things in this way x f g check out this thing i whipped up after school today the diagram on top does x f and the diagram on the bottom does f g the diagram on the right does both it shows how two initial inputs in this case 3 and 3 001 change as they go through the functions f and g at the very bottom you see the heart of this it has δg δf times δf δx δg δx end of sam shah s bit proof of the chain rule we can be more exact and use the derivatives and show that the formula is true for y g f x with f x x n and g f f m the direct solution for y is the derivative of x mn that is mnx mn 1 the formula solution is mf m 1 times nx n 1 giving mnx n m 1 times x n 1 and finally mnx mn 1 some definitions a function is a process which converts an input into a corresponding output in symbols input f output examples the input is transformed converted into the output usually by a formula or an expression using the values of the input output 2 input 5 or you can write this as output 2 times input 5 the input and the output are both expressions which can be a a number or b a single variable x or y or z or c a more complicated expression the commonest form for the input output relationship for a function f is as an example f x 3x 4 where x is an input and the corresponding output is 3x 4 f is the label of the function and f x is the expression whose value is 3x 4 f x 3x 4 is then an equation the equation can be seen for example as f 8 3 x 8 4 or f y 3y 4 using y as the input or f z 2 5z 7 3 z 2 5z 7 4 using an expression in its most simple formulation the input is not present and the equation is simply f 3y 4 where the input is identified as the y now if an equation has a single variable on the left and an expression on the right then a it can be interpreted as a function functional form with f y 3y 4 and b the expression on the right can be substituted for the variable on the left example let g a a 2 be a function g with output a 2 then it can be identified with the equation g a 2 let g have the input x 2 4x 3 then the output is x 2 2 4 x 2 x 3 which is x 2 1 surprise surprise leave a comment filed under uncategorized february 17 2018 3 16 am the chain rule and the theory i found this account of the chain rule from sam shah https samjshah com 2018 02 08 there might be light at the end of the chain rule tunnel maybe the partial text is as follows find two points close to each other like x g f x and x 0 001 g f x 0 001 find the slope between those two points g f x 0 001 g f x x 0 001 x there we go an approximation for the derivative we can use limits to write the exact expression for the derivative if we want but that doesn t help us understand that d dx g f x g f x f x on any level they seem disconnected but i m on my way there i m following things in this way x f g check out this thing i whipped up after school today the diagram on top does x rightarrow f and the diagram on the bottom does f rightarrow g the diagram on the right does both it shows how two initial inputs in this case 3 and 3 001 change as they go through the functions f and g at the very bottom you see the heart of this it has δg δf times δf δx δg δx end of sam shah s bit we can be more exact and use the derivatives and show that the formula is true for y g f x with f x x n and g f f m the direct solution for y is the derivative of x mn that is mnx mn 1 the formula solution is mf m 1 times nx n 1 giving mnx n m 1 times x n 1 and finally mnx mn 1 some definitions a function is a process which converts an input into a corresponding output in symbols input f output examples the input is transformed converted into the output usually by a formula or an expression using the values of the input output 2 input 5 or you can write this as output 2 x input 5 the input and the output are both expressions which can be a a number or b a single variable x or y or z or c a more complicated expression the commonest form for the input output relationship for a function f is as an example f x 3x 4 where x is an input and the corresponding output is 3x 4 f is the label of the function and f x is the expression whose value is 3x 4 f x 3x 4 is then an equation the equation can be seen for example as f 8 3 x 8 4 or f y 3y 4 using y as the input or f z 2 5z 7 3 z 2 5z 7 4 using an expression in its most simple formulation the input is not present and the equation is simply f 3y 4 where the input is identified as the y now if an equation has a single variable on the left and an expression on the right then a it can be interpreted as a function functional form with f y 3y 4 and b the expression on the right can be substituted for the variable on the left example let g a a 2 be a function g with output a 2 then it can be identified with the equation g a 2 let g have the input x 2 4x 3 then the output is x 2 2 4 x 2 x 3 which is x 2 1 surprise surprise 2 comments filed under uncategorized older posts search for recent posts comments 100and 101 division of fractions the 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