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FOURTH GRADE / MATHEMATICS / NATURAL NUMBERS 1.3

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

 In mathematics, natural numbers are

    ♦ 
the ordinary counting numbers 1, 2, 3, ...

    ♦ sometimes zero is also included.


    Ref.-1

    Ref.-2
y
1. Given is the equation KLM - M5 = 350 - 11 where K, L and M stand for digits.
What should the sum of these digits be ?


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RY2
RY2



a) 10
b) 15
c) 19
d) 25


Calculate first 350-11 and then see the problem as given in ry1 .
M should be 4, since 14-9 = 5 is the sole possibility for the ones.
After you fibd the value of M subtrahend is known (45). Just add it to the difference to find 45+339 = 384 .
2. The outcome of (A+16)/A is desired to be a natural number. How many natural numbers A can be found that satisfy this demand ?

a) 5
b) 7
c) 10
d) 16
Write the given formula in the form : 1 + 16/A .
In order to have 16/A a natural number A may be 1, 2 , 4 , and ... .
3. How much does the number 37875 increase if the places of 7 and 8 are exchanged ?

a) 700
b) 900
c) 1800
d) 2700
The number should increase because of this exchange .
4. From the numbers having a digit sum of 21 the greatest and the least are chosen. What is their difference ?

a) 1819
b) 426
c) 111
d) 594
One of the numbers is 993.
5. Which number is the greatest of all ?

a) 86 400 000
b) (8.3+86 400)x365
c) 8.3x86 400x365
d) (86 400/8.3)x365
If thousand or 8.3 x 365 geater ...
6. Two natural numbers both being a multiple of 3 have a sum of 27
How many possible cases exist that satisfy these conditions ?

a) 2
b) 3
c) 4
d) 5
18 and 9 ; 12 and 15 ; etc.
7. Numbers we count with we call natural numbers but in a sense they are not exactly natural. In nature no two objects are exactly equal.
Considering this fact we may want to give a different name to all numbers as we count forward and make all of them distict without a need for coding.
What could be the number of these numbers, coded in our base ten coding ?

a) 10
b) 1 000
c) 100 000
d) 1 000 000
We have to teach these names to next generations and this should take a plausible amount of time, say a year or a couple of years.
y
8. Counting on Oxford Dictionaries we may assume there are 250 000 distinct words in English.
Suppose we make twofold use of them and we use each word's order number as a counting number (natural number) and we understand somehow in which sense the word is being used.
For the binary operation addition we would need some form like "this many and this many" without using new words but subtraction would also be interesting.
How many subtraction results would we need to know by heart ?



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.

RY1 RY2


a) 250 000
b) The sum of all numbers from 1 to 250 000
c) 249 999
d) The sum of all numbers from 1 to 249 999
It is just like knowing all possible distances between N consecutively equidistant points on a line.
As far as the absolute values is concerned we have n-1 absolute values.
But we have N-1 + N-2 + ...+ N-(N-1) distinct distances.
9. How many subtraction results we need to know by heart in case of base 10 coding ?

a) 153
b) 72
c) 18
d) 9


We need to know 18-9 ; 17-9 and 17-8 ; 16-9, 16-8 and 16-6 ; 15-9, 15-8, 15-7, 15-6 ; ... ;11-9, 11-8, 11-7, 11-6, 11-5, 11-4, 11-3, 11-2 which makes a total of 36.
And of course 9-8, ... , 9-1 ; 8-7, ... , 8-1 ; ... ; 2-1 which gives another 36 .
10. Which statement below, referring base ten coding and considering our mental abilities, is true ?

a) It is nonsense
b) It is the best
c) It belongs to one of the most appropriates
d) It is hard to learn
Base 8 and 12 may well compete with it.
11. Considering queations 7 to 10 we learn many important issues about our education; we see that we learn many things without thinking on it very much .
Which of the followings may have an impact on this fact ?

I- Our education system is based on our basic needs.
II- There exist an huge amount of knowledge gathered on the long way behind us.
III- Our system is aiming education of masses not individuals.

Note: Well I am aware that this question is not a mathemetical one, but I think it is of extreme importance; especially nowadays.

a) all
b) I and II
c) I and III
d) II and III
Rather say economic developement and growth .
Huge amount and large audience makes concentration on operative teaching a must. We may say details may be learned as expertise is gained which is quite true, but sufficient global cooperation between experts of all fields fails and is almost impossible.
12. Which term describes the coding given in questions 7 and 8 best ?

a) base 1 coding
b) quasi-baseless coding
c) baseless coding
d) base infinity coding
Base 1 coding is impossible, we can not code all numbers by using only one number/sign. We could not determine its place, its relative place.
We can of course draw a bar corresponding to every object but it is then a picture without coding.
If drawing a picture is defined and accepted as coding, then such a situation may also be called baseless coding. It will be extremely hard to find enogh place as the numbers grow.
13. Another extreme case is base 2 coding [(Aykhan's coding) Ref./ Questions 7-8.]  There we have only two signs, two numbers. One of them indicates existence and the other non-existence.
It is also called binary coding, all computers hardware can work only with this coding.



How could we code 72 (decimally coded) in this way of coding ?

a) 1001000
b) 1100000
c) 1000001
d) 1010100
16 = 10000 , 32 = 100000 and 64 = 1000000 (underlined means decimally coded) .
14. In case of quasi baseless coding we had difficulties with inverse operations (subtraction and division). What is the difficulty of binary coding ?

a) numbers are too long even if they are relatively small.
b) no relative difficulty.
c) multiplication by 2 is more difficult.
d) division by 2 is difficult.
15. Since our quasi baseless coding is not base 250 000 coding we would not use new terms for numbers greater than 250 000 and we will continue to use (with some additional words) the old terms instead.
Which of the following statements is/are true ?

I- During addition we should say something like "this much and this much" calculate the result and express in our terms.
II- During multiplication we should say something like "this much times this much" calculate the result and express in our terms.
III- Division would be a burden, a real burden.

a) Only I
b) I and II
c) II and III
d) all
y
16. Since our memory can not store all the necessary data of our table of subtractions in quasi baseless coding we would need some tables, some look up tables. There we could divide the data into books and and pages having equal numbers of data (for only one fixed term vs. all the other) and this would biring us closer to base-x coding. Assuming the fixed term would be the last term, how many books, pages and terms per page respectively could form a plausible distribution ?

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RY1 RY2



a) 20 , 125 , 100
b) 25 , 100 , 100
c) 63 , 63 , 63
d) 10 , 250 , 100
having equal numbers of data => 63x63x63 = 250 047 .
17. Assuming an average life of 80 years, how many seconds are there in our average life ?

a) 2 522 880 000 000
b) 100 000
c) 2 522 880
d) 2 522 880 000
18. The fastest periodic phenomena in our universe may roughly be assumed to repeat itself 1 000 000 000 000 000 000 000 000 000 times in a second. How many times does it repeat in an average human life ?

a) 2 522 880 000 000 000 000 000 000 000 000 000 000
b) 22 522 880 000 000 000 000 000 000 000 000 000
c) 2 522 880 000 000 000 000 000 000 000 000 000 000
d) 25 522 880 000 000 000 000 000 000 000 000 000
19. The number that is greater than every number is named as "infinite". The ratio of the answer of question 18 to this concept warns us to be careful as we use it .
How big is this ratio ?

a) 0
b) 1
c) infinite
d) 10
Infinity and a point, a real 0 on any axis are just abstractions/idealizations and not realities. We should be extremely careful as we use them.
20. Which number(s) has/have the same value in all possible codings ?

a) 0
b) 1
c) 0 and 1
d) 0 and infinite
We can not show any physical example of 0 but 0 is a number by our definitions, infinite is not a number.
21. An interesting example of understanding how fast numbers grow is folding a piece of paper 20 times. We think we can manage it.
How thick (long) will a 0.5 mm. thick paper get if it is folded 20 times ?

a) 10 000 mm (10 m.)
b) 1 000 mm. (about 1 m.)
c) 248 576 mm (almost a quarter km.)
d) 524 288 mm ( more than 524 m.)
Try and see how many times you can fold a piece of paper, a page of your note book.
22. The unordered and ordered partitions of 4 is seen in the figure below.

!

How many unordered patitions should the natuaral number 5 have ?

a) 6
b) 7
c) 8
d) 9
Better write it down as seen on the figure .
23. An example of historical number coding is Roman Numbers
What is the decimal coding of Roman MDCLXXXIX ?

a) 1789
b) 1685
c) 1339
d) 1341
M=1000 , D=500 , C=100 etc. .
24. Which of the following statements are false ?

I- Multiplication and division with Roman Numerals are hard to perform.
II- Addition and subtraction with Roman Numerals are hard to perform.
III- Use of place values are made.

a) I and II
b) II and III
c) Only II
d) I and III
25. In a country that has 5 cents , 10 cents , 25 cents and 50 cents as coins, Paul has 50 cents .
How many possible combination of coins exist for this case and what kind of partitioning will this be ?

a) 4 , unordered
b) 9 , ordered
c) 6 , unordered
d) 5 , unordered
In this case a 50 cent coin is also possible.
One possible combination is 2x25 cents, and another combination may be 25 cents, 10 cents, 10 cents and 5 cents.
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