- Given f (x) = 3x 2 – x + 4, find the simplified form of the following expression, and evaluate at h = 0: This isn't really a functions-operations question, but something like this often arises in the functions-operations context.
- Write x 2-5x as x^2-5.x. Use paranthesis while performing arithmetic operations. Write sinx+cosx+tanx as sin(x)+cos(x)+tan(x) 2. Write secx.tanx as sec(x).tan(x) 3. Write tanx/sinx as tan(x)/sin(x) 4. Use inv to specify inverse and ln to specify natural log respectively Eg:1.
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Main page for digit sum arithmetic
withModular 9 Arithmetic |
Modular 9 arithmetic is the arithmetic of the remainders after division by 9.For example, the remainder for 12 after division by 9 is 3. This is expressed as
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12 = 3 (mod 9)
Likewise 25=7 (mod 9) and 9=0 (mod 9). Note that 12+25=37 and that 37=1 (mod 9).But 3+7=10=1 (mod 9) so the equivalent of the sum of two numbers modulo 9 isequal to the modulo 9 equivalent of the sum of their modulo 9 equivalents.
The modulo 9 equivalent of 12 is 3 which is also the digit sum of 12. This is no coincidence. There is a very close relationship between the modulo 9 equivalentsof numbers and their digit sums.
Integer | Digit Sum | Remainder Function Mod 9 | |||||||||||||||
Multiplication Table for Modulo 9 Arithmetic | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||||||||
0 | 2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | |||||||||
0 | 3 | 6 | 0 | 3 | 6 | 0 | 3 | 6 | |||||||||
0 | 4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 | |||||||||
0 | 5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 | |||||||||
0 | 6 | 3 | 0 | 6 | 3 | 0 | 6 | 3 | |||||||||
0 | 7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 | |||||||||
0 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
If the 0's were replaced by 9's and the table rearranged so the first column becomes the last column and the first rowbecomes the last row the result would be identical to the table for the sequences of digit sums.
The Rearrangement of the Multiplication Table for Modulo 9 Arithmetic With 9 Substituted for 0 | ||||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | 9 |
3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 |
4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 | 9 |
5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 | 9 |
6 | 3 | 9 | 6 | 3 | 9 | 6 | 3 | 9 |
7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 | 9 |
8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 9 |
9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
This table is identical with the table of digit sums for multiples of numbers. The conclusion is that digit sum arithmetic is the virtually the same as modular 9arithmetic except there is a replacement of 0's with 9's. The equivalence of 9 and 0 takes care of a smallproblem. The digit sum of all multiples of 9 is 9 except for the case of 0 times 9 which has a digit sumof 0. So the digit sum of all multiples of 9 is equivalent to 9.
In mathematical terminology thereis an isomorphism between digit sum arithmetic and modular 9 arithmetic. All ofthe properties of associativity, distributivity, communtivity, identities andadditive inverses carry over from modular 9 arithmentic to digit sum arithmetic. Multiplicativeinverses exist for some elements but not all so neither entity is a mathematical field.In the table below the cases of products that yield the multiplicative identity; i.e., 1; are shown in red.
Multiplication Table for Modulo 9 Arithmetic | |||||||||
---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
2 | 0 | 2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 |
3 | 0 | 3 | 6 | 0 | 3 | 6 | 0 | 3 | 6 |
4 | 0 | 4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 |
5 | 0 | 5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 |
6 | 0 | 6 | 3 | 0 | 6 | 3 | 0 | 6 | 3 |
7 | 0 | 7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 |
8 | 0 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
As can be seen from the above table inverses exist for 1, 2, 4, 5, 7 and 8 but not for 0, 3 and 6.
How To Measure Pan Sizes:
Goodtask 4 9 5 X 9 Plywood Sheets
Measure Dimensions: Moviesherlock 6 0 8 mm. To determine the pan’s dimensions always measure inside edge to inside edge of the pan so that you do not include the thickness of the pan in your measurement.
Measure Depth: To measure the depth, place your ruler straight up from the bottom of the pan (do not slant the ruler). If the pan edge is slanted, do not slant the ruler, measure straight up.
Pan Volume: To determine the pan’s volume (how much batter it will hold), pour premeasured water by the cupful until the pan is filled to the brim. Use a liquid measuring cup to pour water into the pan until it reaches the top.
If the new pan makes the batter shallower than in the original recipe, this will cause the heat to reach the center of the pan more quickly and you will have more evaporation. To solve this problem you need to shorten the baking time and raise the temperature of the oven slightly. To substitute a pan that is shallower than the pan in the recipe, reduce the baking time by 1/4.
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If the new pan makes the batter deeper than in the original recipe, this will cause less evaporation and the batter will take longer to cook. To solve this problem you need to lengthen the baking time and lower the temperature of the oven slightly. To substitute a pan that is deeper than the pan in the recipe, increase the baking time by 1/4.
How To Substitute Pans:
The following table will help determine substitutions of pans and dishes of similar approximate size if you do not have the specific sized baking pan, dish, or mold called for in a recipe.
To substitute with glass pans, reduce the baking temperature by 25 degrees.
Pan Size | Pan Substitution |
---|---|
3-cup Baking Dish or Pan: | 8' x 1-1/4 round pan |
4-cup Baking Dish or Pan: | 1 (8') round cake pan 7-1/2' x 3' bundt tube pan 8' x 8' x 2' square pan 8 1/2' x 4-1/2' x 2-1/2' loaf pan 9' x 1-1/2' round layer cake pan 9' x 2' round pie plate (deep dish) 9' x 9' x 1-1/2' rectangular pan 10' x 1-1/2' round pie plate 11' x 7' x 2' rectangular pan |
7-cup Baking dish or Pan: | 8' x 2' round cake pan 9' x 9' x 2' rectangular pan |
8-cup Baking dish or Pan: | 8' x 8' x 2' square pan 9' x 2' round cake pan 9' x 5' x 3' loaf pan 9' x 9' x 1-1/2' square pan 9-1/4' x 2-3/4' ring mold 9-1/2' x 3-1/4' brioche pan 11' x 7' x 1-1/2' baking pan |
9-cup Baking Dish or Pan: | 8' x 3' bundt pan 9' x 3' tube pan |
10-cup Baking Dish or Pan: | 8' x 2-1/2' springform pan 9' x 9' x 2' square pan 11-3/4' x 7-1/2' x 1 3/4' baking pan 13' x 9' x 2' rectangular pan 15-1/2' x 10-1/2' x 1' jelly-roll pan |
11-cup Baking Dish or Pan: | 9' x 3' springform pan 10' x 2' round cake pan |
12-cup Baking Dish or Pan: | 2 (9') round cake pans 9' x 3' angel-cake pan or tube pan 10' x 2-1/2' springform pan 10' x 3-1/2' bundt pan 13' x 9' x 2' metal baking pan 14' x 10-1/2' x 2-1/2' roasting pan |
15-cup Baking dish or Pan: | 13' x 9' x 2' rectangular pan |
16-cup Baking dish or Pan: | 9' x 3-1/2' springform pan 10' x 4' fancy tube mold |
18-cup Baking dish or Pan: | 10' x 4' angel-cake or tube pan |
9-5 Chic
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