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Lesson 15
§12. Arithmetic operations in positional number systems

Arithmetic operations in positional number systems

Arithmetic operations in positional number systems with a base q follow the rules, similar to the rules that are found in the tenth system of numbers.

At the beginning of the school, for the education of children, there is a vicor table of folding and multiplication. Similar tables can be put together, whether it be a positional number system.

12.1. Addition of numbers in the system of numbers with the substava q

Take a look at the appendices of the supplementary table in the ternary (Table 3.2), sixteenth (Table 3.4) and sixteenth (Table 3.3) number systems.

Table 3.2

Addition in the ternary number system

Table 3.3

Addition in the sixteenth system of numbers

Table 3.4

Addition to the highest system of numbers

q take the sum S two numbers BUTі B, you need to add the numbers that they are approved, for the ranks i left to right:

Yakscho a i + b i< q, то s i = a i + b i , старший (i + 1)-й разряд не изменяется;
if a i + b i q, then s i \u003d a i + b i - q, the senior (i + 1)th digit is increased by 1.

Apply:

12.2. Vіdnіmannya of numbers in the system of numbers with subdstava q

Schob at the system number with the basis q take away the margin R two numbers BUTі AT, the requirement is to calculate the differences and approve their digits for the ranks i left to right:

If a i ≥ b i , then r i = a i - b i the senior (i + 1) digit is not changed;
yakscho a i< b i , то r i = a i - b i + g, старший (i + 1)-й разряд уменьшается на 1 (выполняется заём в старшем разряде).

The folding of that visualization of numbers in any positional system of numbers is numbered bit by bit. For znakhodzhennya sumi, units of one and the same category are added up, starting with units of the first category (right-handed). As soon as the sum is one of the order, which is added, the number is changed, equal to the substations of the system, then the sum of the sum is seen as the one of the senior order, and it is added to the judicial order of evil. This addition can be carried out without interruption, like in the tenth system, in the "stovpchik", vikoristovuyuchi table of folding single-digit numbers.

For example, in a system of numbers with a support 4, the folding table may look like this:

An even simpler table is the addition in the dual number system:

Vidnimannya vykonuєmo just like that, like in the tenth system: signing the change of numbers and troubling the numbers in the ranks, starting from the first. It’s impossible to see alone at the rank, “borrow” the loneliness from the greater rank and we can remake them from the loneliness of the court right rank.

How do we add up to the tenth system of numbers?

Let's think about those, how we add up the numbers already in the same way to us, in the tenth.

Naygolovnіshe varto ozumіti razryad. Guess the alphabet of the skin SS, and then it will become easier for you.

Additions in the two system are not affected by anything in the additions in the tenth system. It’s a smut to remember that the alphabet can only take two numbers: 0 and 1. Since we add 1 + 1, then we take 0, and we increase the number by 1 digit. Look at the butt more:

1. We begin to fold yak and called right-handed in bulk. 0 + 0 \u003d 0, also, we write down 0. We pass to the offensive discharge.
2. We add 1 + 1 and take 2, but 2 is not possible in the dual number system, and therefore it is written 0, and 1 is added to the offensive category.
3. We have three singles to go out of the row, 1 + 1 + 1 = 3, these numbers can’t be like that. Mean 3 - 2 \u003d 1. І 1 is added to the offensive discharge.
4. We again have 1 + 1 = 2. We already know that 2 cannot be, so we write 0, and 1 is added to the offensive category.
5. Store more than nothing, then, at a minimum: 10100.

One butt was taken apart by me, the other was done independently:

So, just like in any other systems of numbers, it is necessary to remember the Alphabet. Let's try to fold the viraz.

1. Everything is as it should be, we begin to fold it in bulk on the right. 4+3=7.
2. 5 + 4 \u003d 9. Nine buti cannot, so from 9 we can see 8, we can take 1. And 1 more can be added to the offensive rank.
3. 3 + 7 + 1 \u003d 11. From 11 we see 8, we take 3. I add one to the offensive rank.
4. 6 + 1 = 7.
5. Put away nothing. ID: 7317.

And now try to add on your own:

1. Vikonuyemo already know we don’t forget about the alphabet. 2+1=3.
2. 5+9=14. Guessing Alphabet: 14=E.
3. C \u003d 12. 12 + 8 \u003d 20. Twenty is not in the sixteenth system of numbers. Mean іz 20 vіdnіmaєmo 16 i otrimuієmo 4. І one is added to the offensive rank.
4. 1 + 1 = 2.
5. Put nothing more. Response: 24Е3.

Rehabilitation of number systems

Guess what, like mi tse robimo in the tenth system of numbers.

1. We repair the levoruch, from the smallest to the largest. 2 - 1 = 1.
2. 1 – 0 = 1.
3. 3 - 9 =? Three mensha for nine, that one is comparable to one of the senior rank. 13 - 9 = 4.
4. From the rest of the order, we took one in front of the row, to that 4 - 1 \u003d 3.
5. ID: 3411.

1. Let's fix it like a zavzhd. 1 - 1 = 0.
2. 1 – 0 = 1.
3. Vіd 0 it is not possible to select one. To that we will take one category from the elder. 2 - 1 = 1.
4. Response: 110.

And now you sing on your own:

1. Nothing new, smut - remember the alphabet. 4 - 3 = 1.
2. 5 – 0 = 5.
3. Vіd 3 vіdіbrati 7 mi vіdrazu cannot, for which we need to place one at the senior level. 11 - 7 = 4.
4. Remember that you posted one earlier, 6 - 1 = 5.
5. ID: 5451.

Take the forward butt, and wonder what the result of the sixteenth system will be. Such a chi іnshiy?

1. 4 – 3 = 1.
2. 5 – 0 = 5.
3. Vіd 3 vіdіbrati 7 mi vіdrazu cannot, for which we need to place one at the senior level. 19 - 7 \u003d 12. In the sixteenth system, 12 \u003d C.
4. Remember that you posted one earlier, 6 - 1 = 5
5. Type: 5С51

An example for an independent vision:

Reproduction in number systems

Let's remember once and for all that multiplying in any system of numbers by one will always give the same number.

1. The skin category is multiplied by one, as it is obviously right-handed, and the number 6748 is taken;
2. 6748 is multiplied by 8 and we take the number 53984;
3. We take the operation of multiplying 6748 by 3. We take the number 20244;
4. We add up all 3 numbers, according to the rules. Take 2570988;
5. ID: 2570988.

In a double system, it is even easier to multiply. Mi zavzhdi multiply either by 0 or by one. Golovne, put it respectfully. Let's try.

1. 1101 is multiplied by one, as you sound right-handed in cash, and the number 1101 is taken;
2. Vikonuemo tsyu operation sche 2 more times;
3. We fold all 3 numbers respectfully, remembering the alphabet, not forgetting about the drabinka;
4. ID: 1011011.

An example for an independent vision:

1. 5 x 4 \u003d 20. A 20 \u003d 2 x 8 + 4. The excess in the gap is written down to the number - it will be 4, and 2 will be kept in mind. We try this procedure on the right side and take the number 40234;
2. When multiplying by 0, we take chotiri 0;
3. When multiplied by 7, we take the number 55164;
4. Now we add up the numbers that are taken - 5556634;
5. ID: 5556634.

An example for an independent vision:

All yak zavzhd, smut guess the abetka. Literal numerals, for clarity, translate from the number system for yourself, as multiply, translate back from the letter value.

Let's for accuracy, let's take the multiplication of the 5th number 20A4.

1. 5 x 4 \u003d 20. And 20 \u003d 16 + 4. The excess in the gap is written down to the number - it will be 4, and 1 will be kept in mind.
2. A x 5 + 1 \u003d 10 x 5 + 1 \u003d 51. 51 \u003d 16 x 3 + 3. The excess in the division is written down to the number - it will be 3, and 3 is three in the mind.
3. When multiplying by 0, we take 0 + 3 = 3;
4. 2 x 5 = 10 = A; The result is A334; Vikonuemo tsyu procedure with two smaller numbers;
5. Remember the rule of multiplication by 1;
6. When multiplying by, we have the number 1670С;
7. Now we add up the numbers that are taken - 169B974;
8. ID: 169B974.

An example of an independent solution.

Apply the translation of numbers from different number systems

Butt #1
Transferring the number 12 from the tenth to the two number system
Solution

Let's transfer the number 12 10 to the 2-ichnu system of numbers, for the additional successive subdivision to 2, the docks will not be equal to zero more privately. As a result, the number of leftovers will be taken away from the right-handed written down.

Butt #2
Transferring the number 12.3 from the tenth to the two number system

12.3 10 = 1100.010011001100110011001100110011 2

Let's transfer the number of parts on the 12th of 12.3 10 to the 2-ichnu number system, for the additional successive subdivision to 2, the docks will not be more private than zero. As a result, the number of leftovers will be taken away from the right-handed written down.

We transfer the fractional part 0.3 of the number 12.3 10 to the 2-ichnu system of numbers, for the additional successive multiplication by 2, doti, until the fractional part of the creation does not see zero, otherwise the necessary number of signs after Komi will not be reached. As a result of multiplying the integer part is not equal to zero, it is necessary to replace the value of the integer part by zero. As a result, the number of whole parts of creations will be taken away, written evil to the right.

Butt #3
Transferring the number 10011 from the two system to the tenth number system
Solution

We transfer the number 10011 2 to the tenth number system, for which we write the position of the skin number in the right number, starting from zero

The skin position of the number will be a step of the number 2, but the number system is 2-a. It is necessary to sequentially multiply the skin number 10011 2 by 2 steps of the second position of the number and then add the next step of the second position with the next addition of the step of the second position.

10011 2 = 1 ⋅ 2 4 + 0 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 = 19 10

Butt #4
Transferring the number 11.101 from the two system to the tenth number system

11.101 2 = 3.625 10

We translate the number 11.101 2 from the tenth number system, for which we write the position of the skin number in the number

The skin position of the number will be a step of the number 2, but the number system is 2-a. It is necessary to sequentially multiply the skin number 11.101 2 by 2 steps of the second position of the number and then add the next step of the second position with the next addition of the step of the second position.

11.101 2 = 1 ⋅ 2 1 + 1 ⋅ 2 0 + 1 ⋅ 2 -1 + 0 ⋅ 2 -2 + 1 ⋅ 2 -3 = 3.625 10

Butt #5
Translating the number 1583 s tenth system the sixteenth system of numbers

1583 10 = 62F 16

Let's transfer the number 1583 10 to the 16th number system, for the additional successive subdivision to 16, the docks will not be more private than zero. As a result, the number of leftovers will be taken away from the right-handed written down.

Butt #6
Transferring the number 1583.56 from the tenth system to the sixteenth number system

1583.56 10 = 62F.8F5C28F5C28F5C28F5C28F5C28F5C2 16

Let's transfer the integer part 1583 of the number 1583.56 10 to the 16-digit number system, for the additional successive subdivision to 16, the docks will not be equal to zero. As a result, the number of leftovers will be taken away from the right-handed written down.

Let's transfer the fractional part 0.56 of the number 1583.56 10 to the 16-ary system of numbers, for the additional successive multiplication by 16, doti, until you see zero in the fractional part of the creation, otherwise the necessary number of signs after the Komi will not be reached. As a result of multiplying the integer part is not equal to zero, it is necessary to replace the value of the integer part by zero. As a result, the number of whole parts of creations will be taken away, written evil to the right.

Butt #7
Transferring the number A12DCF from the sixteenth system to the tenth number system

A12DCF 16 = 10563023 10

We transfer the number A12DCF 16 to the tenth number system, for which we write the position of the skin digit in the right number, starting from zero

The skin position of the number will be a step of the number 16, but the system is 16-number. It is necessary to sequentially multiply the skin number A12DCF 16 by the 16th step of the second position of the number and then add the step of the second position with the next supplement of the next step of the second position.
2

A12DCF.12A 16 = 10 ⋅ 16 5 + 1 ⋅ 16 4 + 2 ⋅ 16 3 + 13 ⋅ 16 2 + 12 ⋅ 16 1 + 15 ⋅ 16 0 + 1 ⋅ 1

1010100011 2 = 1 ⋅ 2 9 + 0 ⋅ 2 8 + 1 ⋅ 2 7 + 0 ⋅ 2 6 + 1 ⋅ 2 5 + 0 ⋅ 2 4 + 0 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 = 675 10

Let's transfer the number 675 10 to the 16th number system, for the help of the successive division by 16, the docks will not be more privately equal to zero. As a result, the number of leftovers will be taken away from the right-handed written down.

The calculator allows you to convert numbers and fractional numbers from one number system to another. Substava of the number system can be less than 2 and more than 36 (10 digits and 26 Latin letters, after all). The number of numbers can be changed up to 30 symbols. Tick ​​the symbol for entering fractional numbers. abo, . To convert a number from one system to another, enter the other number in the first field, base exit system Counting in another is the basis of the number system, it is necessary to translate the number into the yaku, in the third field, after which press the "Remove record" button.

Cob number recorded in 2 3 4 5 6 7 8 9 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 5 number system.

I want to record the number in 2 3 4 5 6 7 8 9 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 number system.

Take the record

Vikonano translation: 3336969

You can also buti tsikavo:

• Truth table calculator. SDNF. SKNF. Polynomial of Zhegalkin

number systems

Number systems are divided into two types: positionalі not positional. We respect the Arab system, it’s positional, and even more Roman - it’s not positional. In positional systems, the position of the digit in the number uniquely indicates the value of the number. It's easy to understand when you look at the examples of such a number.

butt 1. Take the number 5921 from the tenth number system. The number is numbered right-handedly, starting from zero:

The number 5921 can be written in an offensive form: 5921 \u003d 5000 +900 +20 +1 \u003d 5 10 3 +9 10 2 +2 10 1 +1 10 0. The number 10 is a characteristic that defines the number system. In the quality of the steps, the value of the position of the given number is taken.

butt 2. Let's look at the tenth number 1234.567. It is numbered yogo starting from the zero position of the number from the tenth point to the left and to the right:

The number 1234.567 can be written in an offensive form: 1234.567 = 1000+200+30+4+0.5+0.06+0.007 = 1 10 3 +2 10 2 +3 10 1 +4 10 0 +5 10 -1 + 6 10 -2 +7 10 -3.

Converting numbers from one number system to another

Biggest in a simple way converting the number from one number system to another, є converting the number from the beginning to the tenth number system, and then taking the result into the required number system.

Converting numbers from any number system to the tenth number system

To convert a number from a system of numbers to ten, it’s enough to number the first order, starting from zero (the lefthand discharge in the decimal point) similarly to the butts 1 or 2. We know the sum of the creations of the digits of the number on the basis of the number system in the world of the position of the digit of the number:

1. Convert number 1001101.1101 2 to tenth number system.
Solution: 10011.1101 2 = 1 2 4 +0 2 3 +0 2 2 +1 2 1 +1 2 0 +1 2 -1 +1 2 -2 +0 2 -3 +1 2 - 4 = 16 +2 +1 +0.5 +0.25 +0.0625 = 19.8125 10
Suggestion: 10011.1101 2 = 19.8125 10

2. Convert number E8F.2D 16 to tenth number system.
Solution: E8F.2D 16 = 14 16 2 +8 16 1 +15 16 0 +2 16 -1 +13 16 -2 = 3584+128+15+0.125+0.05078125 = 3727.17578125 10
Suggestion: E8F.2D 16 = 3727.17578125 10

Converting numbers from the tenth number system to the other number system

For the transfer of numbers from the tenth system of numbers to the other system of numbers, the number and the fractional part of the number need to be shifted okremo.

Translation of the whole part of a number from the tenth number system to the other number system

The number part is transferred from the tenth number system to the other number system, after an additional successive subdivision of the whole number part on the basis of the number system to the removal of a whole excess, less than the basis of the number system. The result of the transfer will be a record from the leftovers, starting from the rest.

3. Convert number 273 10 to octal number system.
Solution: 273/8 = 34 - excess 1, 34/8 = 4 - excess 2, 4 less than 8, then the calculation is completed. The entry from the leftovers looks like this: 421
Revising: 4 8 2 +2 8 1 +1 8 0 = 256 +16 +1 = 273 = 273 Otzhe translation vikonano correctly.
Suggestion: 273 10 = 421 8

Let's take a look at the translation of the correct decimal fractions in different number systems.

Converting the shot part of a number from the tenth number system to the next number system

Guessing, the correct decimal fraction is called speech number with zero whole part. To translate such a number into a number system with a base N, it is necessary to sequentially multiply the number by N until the fractional part is reset to zero, or the necessary number of discharges is removed. If, when multiplying, the number comes out with a whole part, while seeing zero, then the whole part is far from being protected, so that it is sequentially entered to the result.

4. Convert number 0.125 10 to two number system.
Solution: 0.125 2 = 0.25 (0 is the number of the part, as it will become the first digit of the result), 0.25 2 = 0.5 (0 is another digit of the result), 0.5 2 = 1.0 (1 is the third digit of the result, and so the fractional part is closer to zero , then the translation is complete).
Suggestion: 0.125 10 = 0.001 2