As is known, multiplication of numbers comes down to the summation of partial products obtained by multiplying the current digit of the multiplier IN to the multiplicand L. For binary numbers, partial products are equal to the multiplicand or zero. Therefore, multiplication of binary numbers is reduced to sequential summation of partial products with a shift. For decimal numbers, partial products can take on 10 different values, including zero. Therefore, to obtain partial products, instead of multiplication, multiple sequential summation of the multiplicand L can be used. To illustrate the algorithm for multiplying decimal numbers, we will use an example.
Example 2.26. Pa fig. 2.15, A The multiplication of integer decimal numbers A x b = 54 x 23 is given, starting from the least significant digit of the multiplier. The following algorithm is used for multiplication:
0 is taken as the initial state. The first sum is obtained by adding the multiplicand A = 54 to zero. Then the multiplicand is added to the first sum again A= 54. And finally, after the third summation, the first partial product is obtained, equal to 0 "+ 54 + 54 + 54 = 162;
Rice. 2.15. Algorithm for multiplying integer decimal numbers 54 x 23(A) and the principle of its implementation(b)
- the first partial product is shifted one bit to the right (or the multiplicand to the left);
- the multiplicand is added twice to the highest digits of the first partial product: 16 + 54 + 54 = 124;
- after combining the resulting sum 124 with the least significant 2 of the first partial product, the product 1242 is found.
Let us consider, using an example, the possibility of a circuit implementation of an algorithm using the operations of summation, subtraction and shift.
Example 2.27. Let it be in the register R t the multiplicand is permanently stored A = 54. In the initial state to the register R 2 place the multiplier IN= 23, and register R 3 is loaded with zeros. To obtain the first partial product (162), we add the multiplicand three times to the contents of the register A = 54, decreasing the contents of the register each time by one R T After the least significant bit of the register R., becomes equal to zero, shift the contents of both registers /?. to the right by one bit, and R.,. Presence of 0 in the least significant digit R 2c indicates that the formation of the partial product is complete and a shift needs to be made. Then we perform two operations of adding the multiplicand A= 54 with the contents of the register and subtracting one from the contents of the register R 0. After the second operation, the least significant digit of the register R., will become equal to zero. Therefore, by shifting the contents of the registers to the right by one bit R 3 and R Y we obtain the required product P = 1242.
The implementation of the algorithm for multiplying decimal numbers in binary decimal codes (Fig. 2.16) has features associated with performing addition and subtraction operations
Rice. 2.16.
(see paragraph 2.3), as well as shifting the tetrad by four bits. Let's consider them under the conditions of Example 2.27.
Example 2.28. Multiplying floating point numbers. To obtain the product of numbers A and B c floating point must be defined M c = M l x M n, R With = P{ + R n. In this case, the rules of multiplication and algebraic addition of fixed-point numbers are used. The product is assigned a "+" sign if the multiplicand and the multiplier have the same signs, and a "-" sign if their signs are different. If necessary, the resulting mantissa is normalized with appropriate order correction.
Example 2.29. Multiplying binary normalized numbers:
When performing a multiplication operation, special cases may occur that are handled by special processor instructions. For example, if one of the factors is equal to zero, the multiplication operation is not performed (blocked) and a zero result is immediately generated.
1. An ordinary fraction whose denominator is 10, 100, 1000, etc. is called a decimal fraction.
2. Fractions with a denominator of 10 n can be written as a decimal.
3. If you add one or more zeros to the decimal fraction on the right, you get a fraction equal to the given one.
4. If in a decimal fraction one or more zeros are removed from the right, you will get a fraction equal to the given one.
5. The integer part from the fractional part in the decimal notation of a number is separated by a comma.
6. The fractional part from the integer part in the decimal notation of a number is separated by a comma.
7. A decimal fraction that has a finite number of digits after the decimal point is called a finite decimal fraction.
8. A decimal fraction that has an infinite number of digits after the decimal point is called an infinite decimal fraction.
9. Infinite decimal fractions are divided into periodic and non-periodic decimal fractions
10. A consecutively repeated digit or minimal group of digits in the notation of an infinite decimal fraction after the decimal point is called the period of this infinite decimal fraction.
11. Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 and 5 are written as a final decimal fraction.
12. Irreducible ordinary fractions, in the denominator of which, in addition to 2 and 5, there are other prime factors, are written as an infinite decimal fraction.
13. The rule for converting a decimal fraction into an ordinary fraction.
To write a decimal fraction as a fraction, you need to:
1) leave the whole part unchanged;
2) write the number after the decimal point in the numerator, and in the denominator - one and as many zeros as there are digits after the decimal point in the decimal fraction.
14. The rule for converting a fraction to a decimal.
1) (1 method) In order to write an irreducible ordinary fraction, the denominator of which does not contain other prime factors other than 2 and 5, as a decimal, you need to present it as a fraction with the denominator 10,100,1000, etc.
(2nd method) – divide the numerator by the denominator.
2) In order to write an irreducible ordinary fraction, in the denominator of which, in addition to 2 and 5, there are other prime factors as a decimal, you need to divide the numerator by the denominator.
15. Decimal places –…hundreds, tens, units, tenths, hundredths, thousandths…ten-thousandths….
16. The numbers in the decimal fraction to the right of the decimal point are called decimals.
17. Comparison of decimals:
1) (1st method) On a coordinate ray, the smaller decimal fraction is located to the left, and the larger decimal fraction is located to the right. Equal decimal fractions are represented on the coordinate ray by the same point.
2) (2nd method) Decimal fractions are compared place by digit, starting with the highest digit.
1) If the integer parts of decimal fractions are different, then the greater is the decimal fraction whose integer part is larger, and the lesser is the decimal fraction whose integer part is smaller.
2) if the whole parts of decimal fractions are the same, then the greater is the decimal fraction whose first of the non-matching digits written after the decimal point is greater.
18. Rules for rounding the whole part of a decimal fraction. To round a decimal fraction to the decimal place tens, hundreds, etc., you can discard its fractional part and apply the rule of rounding natural numbers to the learned number.
19. Rules for rounding the fractional part of a decimal. To round a decimal to the units, tenths, hundredths, etc. place, you can:
1) discard all digits following this digit;
2) if the first discarded digit is 5, 6, 7, 8, 9, then increase the resulting number by one digit to which we round;
3) if the first discarded digit is 0,1,2,3,4. then leave the resulting number unchanged.
20. The rule for adding (subtracting) decimal fractions. To add (subtract) decimal fractions, you need to:
1) equalize the number of decimal places in decimal fractions;
2) write them down one after the other so that the comma is under the comma, and the numbers of the same digits are one under the other;
3) perform addition (subtraction) bit by bit;
4) place a comma in the resulting value of the sum (difference) under the commas of the terms (minued and subtracted).
21. The rule for multiplying a decimal fraction by a natural number. To multiply a decimal fraction by a natural number, you need to:
1) multiply it by this number, ignoring the comma;
2) in the resulting product, separate as many digits on the right with a comma as there are in the decimal fraction separated by a comma.
22. The rule for multiplying a decimal fraction by the numbers 10,100,1000, etc. To multiply a decimal fraction by 10,100,1000, etc., you need to move the decimal point to the right by as many digits as there are zeros in the digit unit.
23. The rule for multiplying a decimal fraction by the numbers 0.1; 0.01; 0.01, etc. To multiply a decimal by 0.1; 0.01; 0.01, etc., you need to move the decimal point to the left by as many digits as there are decimal places in the divisor.
24. Rule for multiplying decimals. To multiply decimal fractions:
1) multiply them, ignoring the comma;
2) in the resulting product, separate with a comma as many digits on the right as there are separated by a comma in two factors together.
25. The rule for dividing a decimal fraction by numbers 10,100,1000, etc. To divide a decimal fraction by 10,100,1000, etc., you need to move the decimal point to the left by as many digits as there are zeros in the digit unit.
26. The rule for dividing a decimal fraction by numbers 0.1; 0.01; 0.01, etc. To divide a decimal by 0.1; 0.01; 0.01, etc., you need to move the decimal point to the right by as many digits as there are decimal places in the divisor.
27. The rule for dividing a decimal fraction by a natural number. To divide a decimal fraction by a natural number, you need to:
1) divide it by this number, ignoring the comma; 2) in the resulting quotient, separate as many digits on the right with a comma as there are separated by a comma in the decimal fraction.
28. Dividing a decimal by a decimal. To divide a number by a decimal fraction:
1) in the dividend and divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor;
2) perform division by a natural number.
Comment:
For example, 0.333...=0,(3). They read: “About as many as three in a period.” If in an infinite periodic decimal fraction the period begins immediately after the decimal point, then it is called a pure periodic decimal fraction. If a periodic decimal fraction has other decimal places between the decimal point and the period, it is called a mixed periodic decimal fraction. Integers can be written as pure periodic decimal fractions with a period equal to zero. Infinite decimal non-periodic fractions are called irrational numbers. Irrational numbers are written only as an infinite decimal non-periodic fraction.
The topic of Multiplying Decimals includes multiplying a decimal by a natural number, multiplying a decimal by a decimal, and some important special cases. Let's write down all the rules for this topic on one page.
To multiply a decimal fraction by a natural number, you need
- in the resulting product, separate as many digits after the decimal point as there are after the decimal point in the decimal fraction.
Examples of multiplying a decimal fraction by a natural number.
We multiply without paying attention to the comma, that is, 342∙7=2394. There are two digits after the decimal point in the decimal fraction 3.42. Therefore, in the resulting product we separate two numbers after the decimal point: 23.94.
Thus, 3.42∙7=23.94.
We multiply the numbers, ignoring the comma: 7135∙2=14270. In the resulting result, you should separate the last two digits with a comma: 142.70. Since zeros after the decimal point are not written at the end of the decimal fraction, then
71,35∙2=142,70=142,7.
3) 0, 000836∙17=?
We multiply without taking into account the comma: 836∙17=14212. Since a decimal fraction has 6 digits after the decimal point, the resulting product must also have 6 digits after the decimal point. Since the result is a total of 5 digits, we supplement the missing one digit with a zero. We assign this zero in front of the number: .01412. When receiving such a record, a zero is written before the comma in the integer part: 0.01412.
To multiply two decimal fractions, you need:
- multiply numbers without paying attention to the comma;
- in the resulting product, separate as many digits after the decimal point as there are after the decimal points in both factors together.
Examples of multiplying decimals.
We multiply the numbers without paying attention to the comma: 13∙4=52. In the resulting product, you should write down as many digits after the decimal point as there are after the decimal point in both factors together. In the first factor 1.3 there is one digit after the decimal point, in the second factor 0.4 there is one digit after the decimal point, in total 1+1=2 digits, the result must be separated by a comma: 0.52 (by adding a zero before the decimal point):
2) 3,00504∙0,025=?
We multiply without taking into account the comma: 300504∙25=7512600. In the resulting product, you need to get as many digits after the decimal point as there are in both factors after the decimal point together, that is, 5 + 3 = 8 digits. We supplement the missing number of digits with zero. We discard the zeros after the decimal point at the end of the decimal fraction.
3,00504∙0,025=0,07512600=0,075126.
3) 1,37∙0,0061=?
The product without commas is 137∙61=8357. After the decimal point there should be 2+4=6 digits. We supplement the number of digits missing up to 6 with two zeros (we write them in front of the number 8357. In the first place, before the comma in the integer part, we write a zero:
1,37∙0,0061=0,008357.
3.Special cases of multiplying decimal fractions.
To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma in the fraction notation to 1, 2, 3, 4, etc. digits to the right.
Examples.
Move the comma one digit to the right:
1) 7.9∙10=79 (here 79.=79);
2) 8,53∙10=85,3;
3) 0, 6541=6,541.
Move the comma two digits to the right:
1) 7,04∙100=704;
2) 3,8754∙100=387,54;
3) 4.5∙100=450 (there is only one digit after the decimal point. The missing 1 digit is supplemented with a zero).
Move the comma three digits to the right:
1) 45,8096∙1000=45809,6;
2) 0.67∙1000=670 (there are 2 digits after the decimal point. The missing 1 digit is supplemented with zero);
Mathematical-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percentage calculations and other operations.
Solution:
How to use a math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Entering natural integers, zero. To get a negative integer, you must press the +/- key |
| . | semicolon) | Separator to indicate a decimal fraction. If there is no number before the point (comma), the calculator will automatically substitute a zero before the point. For example: .5 - 0.5 will be written |
| + | plus sign | Adding numbers (integers, decimals) |
| - | minus sign | Subtracting numbers (integers, decimals) |
| ÷ | division sign | Dividing numbers (integers, decimals) |
| X | multiplication sign | Multiplying numbers (integers, decimals) |
| √ | root | Extracting the root of a number. When you press the “root” button again, the root of the result is calculated. For example: root of 16 = 4; root of 4 = 2 |
| x 2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1/x | fraction | Output in decimal fractions. The numerator is 1, the denominator is the entered number |
| % | percent | Getting a percentage of a number. To work, you need to enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
| ( | open parenthesis | An open parenthesis to specify the calculation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
| ) | closed parenthesis | A closed parenthesis to specify the calculation priority. An open parenthesis is required |
| ± | plus minus | Reverses sign |
| = | equals | Displays the result of the solution. Also above the calculator, in the “Solution” field, intermediate calculations and the result are displayed. |
| ← | deleting a character | Removes the last character |
| WITH | reset | Reset button. Completely resets the calculator to position "0" |
Algorithm of the online calculator using examples
Addition.
Addition of natural integers (5 + 7 = 12)
Addition of integer natural and negative numbers ( 5 + (-2) = 3 )
Adding decimal fractions (0.3 + 5.2 = 5.5)
Subtraction.
Subtracting natural integers ( 7 - 5 = 2 )
Subtracting natural and negative integers ( 5 - (-2) = 7 )
Subtracting decimal fractions (6.5 - 1.2 = 4.3)
Multiplication.
Product of natural integers (3 * 7 = 21)
Product of natural and negative integers ( 5 * (-3) = -15 )
Product of decimal fractions ( 0.5 * 0.6 = 0.3 )
Division.
Division of natural integers (27 / 3 = 9)
Division of natural and negative integers (15 / (-3) = -5)
Division of decimal fractions (6.2 / 2 = 3.1)
Extracting the root of a number.
Extracting the root of an integer ( root(9) = 3)
Extracting the root of decimal fractions (root(2.5) = 1.58)
Extracting the root of a sum of numbers ( root(56 + 25) = 9)
Extracting the root of the difference between numbers (root (32 – 7) = 5)
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ((2,2)2 = 4.84)
Conversion to decimal fractions.
Calculating percentages of a number
Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )
18% of the number 140 is (140 * 0.18 = 25.2)
In this article we will look at the action of multiplying decimals. Let's start by stating the general principles, then show how to multiply one decimal fraction by another and consider the method of multiplication by a column. All definitions will be illustrated with examples. Then we will look at how to correctly multiply decimal fractions by ordinary, as well as mixed and natural numbers (including 100, 10, etc.)
In this material, we will only touch on the rules for multiplying positive fractions. Cases with negative numbers are dealt with separately in articles on multiplying rational and real numbers.
Yandex.RTB R-A-339285-1
Let us formulate general principles that must be followed when solving problems involving multiplying decimal fractions.
Let us first remember that decimal fractions are nothing more than a special form of writing ordinary fractions, therefore, the process of multiplying them can be reduced to a similar one for ordinary fractions. This rule works for both finite and infinite fractions: after converting them to ordinary fractions, it is easy to multiply with them according to the rules we have already learned.
Let's see how such problems are solved.
Example 1
Calculate the product of 1.5 and 0.75.
Solution: First, let's replace decimal fractions with ordinary ones. We know that 0.75 is 75/100, and 1.5 is 15/10. We can reduce the fraction and select the whole part. We will write the resulting result 125 1000 as 1, 125.
Answer: 1 , 125 .
We can use the column counting method, just like for natural numbers.
Example 2
Multiply one periodic fraction 0, (3) by another 2, (36).
First, let's reduce the original fractions to ordinary ones. We will get:
0 , (3) = 0 , 3 + 0 , 03 + 0 , 003 + 0 , 003 + . . . = 0 , 3 1 - 0 , 1 = 0 , 3 9 = 3 9 = 1 3 2 , (36) = 2 + 0 , 36 + 0 , 0036 + . . . = 2 + 0 , 36 1 - 0 , 01 = 2 + 36 99 = 2 + 4 11 = 2 4 11 = 26 11
Therefore, 0, (3) · 2, (36) = 1 3 · 26 11 = 26 33.
The resulting ordinary fraction can be converted to decimal form by dividing the numerator by the denominator in a column:
Answer: 0 , (3) · 2 , (36) = 0 , (78) .
If we have infinite non-periodic fractions in the problem statement, then we need to perform preliminary rounding (see the article on rounding numbers if you have forgotten how to do this). After this, you can perform the multiplication action with already rounded decimal fractions. Let's give an example.
Example 3
Calculate the product of 5, 382... and 0, 2.
Solution
In our problem we have an infinite fraction that must first be rounded to hundredths. It turns out that 5.382... ≈ 5.38. It makes no sense to round the second factor to hundredths. Now you can calculate the required product and write down the answer: 5.38 0.2 = 538 100 2 10 = 1 076 1000 = 1.076.
Answer: 5.382…·0.2 ≈ 1.076.
The column counting method can be used not only for natural numbers. If we have decimals, we can multiply them in exactly the same way. Let's derive the rule:
Definition 1
Multiplying decimal fractions by column is performed in 2 steps:
1. Perform column multiplication, not paying attention to commas.
2. Place a decimal point in the final number, separating it with as many digits on the right side as both factors contain decimal places together. If the result is not enough numbers for this, add zeros to the left.
Let's look at examples of such calculations in practice.
Example 4
Multiply the decimals 63, 37 and 0, 12 by columns.
Solution
First, let's multiply numbers, ignoring decimal points.
Now we need to put the comma in the right place. It will separate the four digits on the right side because the sum of the decimals in both factors is 4. There is no need to add zeros, because enough signs:
Answer: 3.37 0.12 = 7.6044.
Example 5
Calculate how much 3.2601 times 0.0254 is.
Solution
We count without commas. We get the following number:
We will put a comma separating 8 digits on the right side, because the original fractions together have 8 decimal places. But our result has only seven digits, and we cannot do without additional zeros:
Answer: 3.2601 · 0.0254 = 0.08280654.
How to multiply a decimal by 0.001, 0.01, 01, etc.
Multiplying decimals by such numbers is common, so it is important to be able to do it quickly and accurately. Let's write down a special rule that we will use for this multiplication:
Definition 2
If we multiply a decimal by 0, 1, 0, 01, etc., we end up with a number similar to the original fraction, with the decimal point moved to the left the required number of places. If there are not enough numbers to transfer, you need to add zeros to the left.
So, to multiply 45, 34 by 0, 1, you need to move the decimal point in the original decimal fraction by one place. We will end up with 4, 534.
Example 6
Multiply 9.4 by 0.0001.
Solution
We will have to move the decimal point four places according to the number of zeros in the second factor, but the numbers in the first factor are not enough for this. We assign the necessary zeros and get that 9.4 · 0.0001 = 0.00094.
Answer: 0 , 00094 .
For infinite decimals we use the same rule. So, for example, 0, (18) · 0, 01 = 0, 00 (18) or 94, 938... · 0, 1 = 9, 4938.... and etc.
The process of such multiplication is no different from the action of multiplying two decimal fractions. It is convenient to use the column multiplication method if the problem statement contains a final decimal fraction. In this case, it is necessary to take into account all the rules that we talked about in the previous paragraph.
Example 7
Calculate how much 15 · 2.27 is.
Solution
Let's multiply the original numbers with a column and separate two commas.
Answer: 15 · 2.27 = 34.05.
If we multiply a periodic decimal fraction by a natural number, we must first change the decimal fraction to an ordinary one.
Example 8
Calculate the product of 0 , (42) and 22 .
Let us reduce the periodic fraction to ordinary form.
0 , (42) = 0 , 42 + 0 , 0042 + 0 , 000042 + . . . = 0 , 42 1 - 0 , 01 = 0 , 42 0 , 99 = 42 99 = 14 33
0, 42 22 = 14 33 22 = 14 22 3 = 28 3 = 9 1 3
We can write the final result in the form of a periodic decimal fraction as 9, (3).
Answer: 0 , (42) 22 = 9 , (3) .
Infinite fractions must first be rounded before calculations.
Example 9
Calculate how much 4 · 2, 145... will be.
Solution
Let's round the original infinite decimal fraction to hundredths. After this we come to multiplying a natural number and a final decimal fraction:
4 2.145… ≈ 4 2.15 = 8.60.
Answer: 4 · 2, 145… ≈ 8, 60.
How to multiply a decimal by 1000, 100, 10, etc.
Multiplying a decimal fraction by 10, 100, etc. is often encountered in problems, so we will analyze this case separately. The basic rule of multiplication is:
Definition 3
To multiply a decimal fraction by 1000, 100, 10, etc., you need to move its decimal point to 3, 2, 1 digits depending on the multiplier and discard the extra zeros on the left. If there are not enough numbers to move the comma, we add as many zeros to the right as we need.
Let's show with an example exactly how to do this.
Example 10
Multiply 100 and 0.0783.
Solution
To do this, we need to move the decimal point by 2 digits to the right. We will end up with 007, 83 The zeros on the left can be discarded and the result written as 7, 38.
Answer: 0.0783 100 = 7.83.
Example 11
Multiply 0.02 by 10 thousand.
Solution: We will move the comma four digits to the right. We don’t have enough signs for this in the original decimal fraction, so we’ll have to add zeros. In this case, three 0 will be enough. The result is 0, 02000, move the comma and get 00200, 0. Ignoring the zeros on the left, we can write the answer as 200.
Answer: 0.02 · 10,000 = 200.
The rule we have given will work the same in the case of infinite decimal fractions, but here you should be very careful about the period of the final fraction, since it is easy to make a mistake in it.
Example 12
Calculate the product of 5.32 (672) times 1,000.
Solution: first of all, we will write the periodic fraction as 5, 32672672672 ..., so the probability of making a mistake will be less. After this, we can move the comma to the required number of characters (three). The result will be 5326, 726726... Let's enclose the period in brackets and write the answer as 5,326, (726).
Answer: 5, 32 (672) · 1,000 = 5,326, (726) .
If the problem conditions contain infinite non-periodic fractions that must be multiplied by ten, one hundred, a thousand, etc., do not forget to round them before multiplying.
To perform multiplication of this type, you need to represent the decimal fraction as an ordinary fraction and then proceed according to the already familiar rules.
Example 13
Multiply 0, 4 by 3 5 6
Solution
First, let's convert the decimal fraction to an ordinary fraction. We have: 0, 4 = 4 10 = 2 5.
We received the answer in the form of a mixed number. You can write it as a periodic fraction 1, 5 (3).
Answer: 1 , 5 (3) .
If an infinite non-periodic fraction is involved in the calculation, you need to round it to a certain number and then multiply it.
Example 14
Calculate the product 3, 5678. . . · 2 3
Solution
We can represent the second factor as 2 3 = 0, 6666…. Next, round both factors to the thousandth place. After this, we will need to calculate the product of two final decimal fractions 3.568 and 0.667. Let's count with a column and get the answer:
The final result must be rounded to thousandths, since it was to this digit that we rounded the original numbers. It turns out that 2.379856 ≈ 2.380.
Answer: 3, 5678. . . · 2 3 ≈ 2, 380
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