Mathematics Arithmetic [2 ed.]

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MATHEMATICS ARITHMETIC

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MATHEMATICAL OBJECTS Mathematical objects are constants. Mathematical objects are the fundamental building blocks of the mathematical journey and as we will continue to explore our ways through various mathematical foundations or procedures or say some formulas, we will regard mathematical objects as our starting point or the origination point of anything. You cannot define a mathematical object as they are constants and you cannot define a constant. So mathematical objects are the constants that we shall observe and carry out our works with it. We start a solution from mathematical objects but as it will be time consuming or not energy efficient, we will remember certain things at times that will help us gaining knowledge or understanding to the subject the most efficient way possible. Our mathematics is memory based which means, you have to remember stuffs. Because just as aforesaid you have to remember even if they say “you should not memorize math.”

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NUMBERS Numbers are the symbolic representations of mathematical objects as it is not always possible to have the mathematical objects while solving a problem in mathematics. This is the connection between mathematical objects and our working by mean of symbolization. Our numeric system consists of digits which themselves are numbers and they also constitute other numbers in their representation. It ranges from 0 to 9. Here are all the numbers that we have, Numbers 0 1 2 3 4 5

6

7

8

9

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This above table must be ingrained into your brain (although it it). It is counted like 1, 1 + 1 = 2, 1 + 1 + 1 = 3, and so on unto 1 + 1 + ⋯ + 1 = 9. That is how we count. We count objects like one, two, three, four, five six, seven, eight, nine, …. This is how it works. And finally, we convert our verbal form to the symbolic one. The higher numbers are constituted of the fundamental digits or numbers that are represented above. They are represented as … , 9, 10, 11, … . All the numbers following the previous one has a difference of 1 mathematical object. Basically, we reuse the digits by moving the one to the leftwards side. But these symbols can only be used while writing or representing then on a paper. How we actually count verbally marks as the follows, Numbers symbolic form 0 1 2 3

Numbers’ verbal form Zero One Two Three

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4 5 6 7 8 9

Four Five Six Seven Eight Nine

The actual way how we carry out our counting is with the creases present in each of our fingers in our left hand’s palm side. We need to memorize the numbers with respect to the locations from 0 to 9. Numbers symbolic form 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Numbers verbal form Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Twenty-one Twenty-two Twenty-three Twenty-four Twenty-five Twenty-six Twenty-seven Twenty-eight Twenty-nine Thirty Thirty-one Thirty-two Thirty-three Thirty-four Thirty-five Thirty-six Thirty-seven Thirty-eight Thirty-nine Forty Forty-one Forty-two Forty-three Forty-four Forty-five Forty-six Forty-seven Forty-eight

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49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

Forty-nine Fifty Fifty-one Fifty-two Fifty-three Fifty-four Fifty-five Fifty-six Fifty-seven Fifty-eight Fifty-nine Sixty Sixty-one Sixty-two Sixty-three Sixty-four Sixty-five Sixty-six Sixty-seven Sixty-eight Sixty-nine Seventy Seventy-one Seventy-two Seventy-three Seventy-four Seventy-five Seventy-six Seventy-seven Seventy-eight Seventy-nine Eighty Eighty-one Eighty-two Eighty-three Eighty-four Eighty-five Eighty-six Eighty-seven Eighty-eight Eighty-nine Ninety Ninety-one Ninety-two Ninety-three Ninety-four Ninety-five Ninety-six Ninety-seven Ninety-eight

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99 100

Ninety-nine Hundred

It is actually a lot easier to remember it than it looks as it is clear that if you remember twenty, it follows “-one”, “two”, …. Thus, we can remember all of them. For addition or subtraction though, we need to remember up to nineteen. For other operations we need higher.

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HAND-PALM REPRESENTATION OF NUMBERS FROM 𝟎 TO 𝟗

After carrying this thing out for years, you should have mastered the idea of what number to its location of creases of your finger (of left hand preferably). This will be also the configuration if you choose to do it in the right hand.

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OPERATIONS WITH NUMBERS ADDITION (+) Here are a few ways how you can add numbers: •

•

•

Classic way The classic way to add numbers is to list the numbers on a sheet of paper and give a brief mathematical object representation correspondingly and finally adding them. This is how it should be done: 1 − 2 − 3 − 4 − 5 − 6 − 7 − 8 − 9 − 10 − 11 − 12 − 13 − 14 − 15 − 16 − 17 − 18 − The lines shown aside are the mathematical objects. Any number consists of its mathematical object (−) and all that that are there above. To apply this method, the only thing that you need to remember is to count from one (1) to nine (9). Just start from a number to begin and keep on counting to where it is to be added by simply moving down. And where you land is your answer. To benchmark this, the safety is the highest but time consumed is also the highest and memorization required is minimal. Although this is the best way to do add one-digit numbers, it is time consuming. Using hand palm The thing you need to memorize: “One (1) Two (2) Three (3) Four (4) Five (5) Six (6) Seven (7) Eight (8) Nine (9) Ten (10) Eleven (11) Twelve (12) Thirteen (13) Fourteen (14) Fifteen (15) Sixteen (16) Seventeen (17) Eighteen (18)”. This should be ingrained into your memory. To benchmark this, compromising with safety a little bit, it packs in with a greater speed and so is a balanced performance and also some memorization is required. We will prefer this. For its working, an illustration is given below. Memory Memorizing all the one-digit additions can really boos up the speed but it is a hazard to safety. 8 + 2 = 10 7 + 3 = 10 6 + 4 = 10 8 + 4 = 12 7 + 4 = 11 6 + 5 = 11 8 + 5 = 13 7 + 5 = 12 8 + 6 = 14 This is all that you need to remember. Other additions are either too easy or they can be evaluated by multiplication tables. This can be quite advanced anyways for a newbie.

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Another way to do this is searching for the nearest 10. This utility can be used only when the sum of digits is greater tan 10. An illustration is below: 1 1 5 7 8 (Search for nearest 10, 8 + 6 = 8 + 2 − 2 + 6 = 10 + 4 = 14) + 1 8 6 (Do that for the rest) _______________________ 7 6 4 But this method is actually not very effective for addition but for subtraction it is very efficient. Rather memorize to increase speed. Here is how you should carry out the operation of addition with an instance:

+

1

1

3

4

4

3

9

9

________________________ 7

4

3

First step is to hold the ‘9’ here at the 0th position of the palm and continue counting up to the 4th position of the finder creases like – *nine*, ten, eleven, twelve, thirteen – and at the 4th position of the finger, whatever verbally reaches, convert it into symbol mathematically and write as shown. And just like that carry on the operation until its completed.

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SUBTRACTION (−) Here are the ways how you can do it: • • •

Classic method Samilarly write out all the numbers. Hand palm method Search for nearest 10.

Here is how you should carry out the operation of subtraction with an instance: 13

-

4

3

12

5

4

2

3

9

9

_________________________ 1

4

3

First step here is that notice that one’s number of the upper number is lesser than the lower one, so we take the effect as such: 542 − 399 = 530 + 12 − 390 − 9 = 530 − 390 + 12 − 9 = 530 − 390 + 3 = ⋯ = 143 This is how we can carry out the operation and we can have a solution. Now you can start from the 0th position of the finger creases as, *nine*, ten, eleven, twelve. And we end up at finger crease position 3rd of the palm and that’s the result of the first one. And as we continue, we end up to conclude the result.

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MULTIPLICATION (×) This operation is carried out when you have to carry out addition of the same number for a certain number of times with itself. Like we say 2 + 2 + 2 + 2 = 2 × 4 To call in simple terms, it is effective addition and we need to,

Furthermore, if a multiplication is caried out many times such as 3 × 3 × 3 × 3 × 3, we can also write it as 35 . These are called exponents. For a greater multiplication we apply the following, 1

2

2

3

2

3

5

5

6

×

______________________________

+

1

1

4

1

0

(6 × 5), (6 × 3) + 3, (6 × 2) + 2

1

7

5

X

(5 × 5), (5 × 3) + 2, (5 × 2) + 1

______________________________________ 1

3

1

6

0

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The idea behind this is that 235 × 56 = 235(50 + 6) = 235 × 50 + 235 × 6 = ⋯ = 13160, as whatever number you take and multiply it with 10, a zero is added at the back of the number.

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DIVISION (÷) This same algorithm is to be carries out for numbers greater of lower than this illustrated one for a division of □□□ … ÷ □□□ … = □□□ … and that each the boxes contain a digit. □

□

□

…

______________________________________ □□□…

)

□

−

□

□

…

Δ

Δ is the closest number less than □ we can get by multiplying □□□… with □. We bring one digit □ after the subtraction has been done.

________________________

−

□

□

Δ Δ is the closest number less than □ □ we can get by

Δ

Δ

multiplying □□□… with □.

________________________

−

□

□

□

Δ

Δ

Δ

The process continues until we have found our result at

________________________

−

We bring one digit □ after the subtraction has been done.

□

□

□

…

…

…

…

…

the top.

For division to be carried out you need to master up the multiplication first. This operation is carried out like this 189 ÷ 2 = (100 + 80 + 9) ÷ 2 we can Division is to be carried out like this, 1

8

9

______________________________ 2

)

3 −

7

8

2 ________________

−

1

7

1

6

________________

−

1

8

1

8

________________ 0

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