Decimal fractions and actions with them. Decimal division and multiplication
The decimal fraction is used when you need to perform actions with non-integer numbers. This may seem irrational. But this kind of numbers greatly facilitates the mathematical operations that must be performed with them. This understanding comes with time, when their record becomes familiar, and reading does not cause difficulties, and the rules of decimal fractions are mastered. Moreover, all actions are repeated already known, which are learned with natural numbers. Just need to remember some features.
Decimal Fraction Definition
The decimal fraction is a special representation of a non-integer number with a denominator divided by 10, and the answer is obtained as a unit and, possibly, with zeros. In other words, if the denominator is 10, 100, 1000, and so on, then it is more convenient to rewrite the number using a comma.Then the whole part will be located before it, and then the fractional part. Moreover, the recording of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the denominator level.
You can illustrate the above with these numbers:
9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.
Reasons for the use of decimal fractions
Mathematicians took decimal fractions for several reasons:
Simplify recording. Such a fraction is located along one line without a dash between the denominator and the numerator, while visibility does not suffer.
Simplicity in comparison. It is enough just to correlate the figures that are in the same positions, while with ordinary fractions you would have to bring them to a common denominator.
Calculators are not designed for the introduction of ordinary fractions, they use decimal notation for all operations.
How to read such numbers?
The answer is simple: just like an ordinary mixed number with a denominator multiple of 10. The only exception is fractions without integer values, then when reading you need to pronounce “zero integers”.
For example, 45/1000 need to pronounce asforty five thousandthswhile 0.045 will sound likezero point forty five thousandth.
A mixed number with an integer part equal to 7 and a fraction of 17/100, which will be written as 7.17, in both cases will be read asseven point seventeen.
The role of discharges in fractions
True to mark the discharge - this is what requires mathematics. Decimal fractions and their value can change significantly if you write the number in the wrong place. However, this was true before.
To read the decimal digits of an integer, you simply need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If in the whole part "tens" sounded, then after the comma it will be already "tenths".
Clearly this can be seen in this table.Decimals Table
How to write mixed number in decimal?
If the denominator is a number equal to 10 or 100, and others, then the question of how to convert a fraction to decimal, is simple. To do this, it is enough to rewrite all its component parts in a different way.This will help such points:
a little aside to write the numerator of the fraction, at this moment the decimal point is on the right, after the last digit;
move the comma to the left, the most important thing here is to correctly count the numbers - you need to move it to as many positions as there are zeros in the denominator;
if they are missing, then the empty positions should be zeros;
the zeros that were at the end of the numerator are no longer needed, and they can be crossed out;
Before the comma to attribute the integer part, if it was not, then there will also be zero.
Attention. You can not cross out the zeros, which were surrounded by other numbers.
How to be in a situation when the number in the denominator is not only from one and zeros, how to convert the fraction to decimal, you can read a little below. This is important information that you should definitely read.
How to convert a fraction to decimal, if the denominator is an arbitrary number?
There are two options here:
When the denominator can be represented as a number that is equal to ten to any degree.
If such an operation can not be done.
How to check it? It is necessary to expand the denominator into factors.If only 2 and 5 are present in the product, then everything is fine, and the fraction can be easily converted into the final decimal. Otherwise, if 3, 7 and other prime numbers appear, the result will be infinite. Such decimal fraction for ease of use in mathematical operations, it is customary to round. This will be discussed below.
Studies how such decimal fractions are obtained, grade 5. Examples here will be very useful.
Let the denominators are numbers: 40, 24 and 75. The decomposition into prime factors for them will be:
In these examples, only the first fraction can be represented as a finite.
Algorithm for converting an ordinary fraction to a final decimal
Check the expansion of the denominator into prime factors and make sure that it will consist of 2 and 5.
Add to these numbers as many as 2 and 5, so that they become an equal number. They will give the value of the additional multiplier.
Multiply the denominator and numerator by this number. The result is an ordinary fraction, under the line of which is 10 to some extent.
Continue to act as described in paragraph, located a little higher.
If in a task these actions are performed with a mixed number, then it must first be represented as an irregular fraction. And then act on the described scenario.
Representation of a common fraction in the form of a rounded decimal
This way of how to convert a fraction to a decimal one will seem even easier to someone. Because there is not a lot of action in it. It is only necessary to divide the value of the numerator by the denominator.
An infinite number of zeros can be assigned to any number with a decimal part to the right of the comma. This property and need to use.
First write the whole part and put a comma after it. If the fraction is correct, then write zero.
It then relies to divide the numerator by the denominator. So that the number of digits they have is the same. That is, add the required number of zeros to the right of the numerator
Perform division in the bar until the required number of digits is typed. For example, if you need to round up to hundredths, then there should be 3 in the answer. In general, there should be one more digits than you need to get in the end.
Record the intermediate answer after the comma and round it off according to the rules.If the last digit is from 0 to 4, then you just need to drop it. And when it is equal to 5-9, then the one before it must be increased by one, discarding the latter.
Return from decimal fraction to ordinary
In mathematics there are problems when it is more convenient to represent decimal fractions in the form of ordinary ones, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.
For this procedure, do the following:
write the integer part, if it is zero, then you do not need to write anything;
draw a line;
above it, write the numbers from the right side, if the first ones are zeros, then they need to be crossed out;
below the line write the unit with as many zeros as the number of decimals in the initial fraction.
This is all you need to do to convert the decimal fraction to ordinary.
What can be done with decimal fractions?
In mathematics, these will be certain actions with decimal fractions that were previously performed for other numbers.
addition and subtraction;
multiplication and division.
The first action, a comparison, is similar to how it was done for natural numbers.To determine which one is greater, one must compare the discharges of the whole part. If they turn out to be equal, then they go to fractional ones and also compare them in order of digits. That number where there will be a big digit in the senior category will be the answer.
Addition and subtraction of decimal fractions
This is perhaps the most simple action. Because they follow the rules for natural numbers.
So, to perform the addition of decimal fractions, they need to be written one above the other, placing commas in a column. With such a record, whole parts appear to the left of the commas, and fractional to the right. And now you need to add the numbers one by one, as it is done with natural numbers, dropping a comma down. It is necessary to begin addition from the smallest category of a fractional part of number. If the right half is not enough numbers, then append zeros.
When subtracting act the same. And here is the rule that describes the ability to take a unit from the senior level. If the decimal fraction has fewer digits than the deductible, then it is simply assigned zeros.
A little more difficult is the case with tasks where you need to perform multiplication and division of decimal fractions.
How to multiply the decimal fraction in different examples?
The rule by which multiplication of decimal fractions by a natural number is made is:
write them in a column, not paying attention to a comma;
multiply as if they were natural;
separate commas with as many digits as there were in the fractional part of the original number.
A special case is an example in which a natural number is 10 to any degree. Then, to get the answer, you just need to move the comma to the right in as many positions as there are zeros in the other multiplier. In other words, when multiplied by 10, the comma shifts by one digit, by 100 - there will be two of them, and so on. If there are not enough digits in the fractional part, then you need to write zeros on empty positions.
The rule that is used when the task needs to multiply the decimal fractions by another same number:
write them under each other, not paying attention to commas;
multiply as if they were natural;
separate commas with as many digits as there were in the fractional parts of both original fractions together.
A special case are the examples in which one of the factors is equal to 0.1 or 0.01 and more.They need to move the comma to the left by the number of digits in the presented multipliers. That is, if multiplied by 0.1, then the comma is shifted by one position.
How to divide the decimal in different tasks?
The division of decimal fractions into a natural number is performed according to the following rule:
write them for division into a bar, as if they were natural;
divide according to the usual rule until the whole part ends;
put a comma in reply;
continue dividing the fractional component until zero is obtained in the remainder;
if necessary, you can assign the required number of zeros.
If the integer part is zero, then it will not be in the answer either.
Separately, there is a division by numbers equal to ten, hundred, and so on. In such tasks, you need to move the comma to the left by the number of zeros in the divider. It happens that there are not enough digits in the integer part, then zeros are used instead. You can see that this operation is similar to multiplying by 0.1 and numbers similar to it.
To perform decimal fractions, you need to use this rule:
turn the divisor into a natural number, and for this move the comma in it to the right to the end;
move the comma and in the dividend by the same number of digits;
act on the previous scenario.
The division by 0.1; 0.01 and other similar numbers. In such examples, the comma is shifted to the right by the number of digits in the fractional part. If they are over, then you need to add the missing number of zeros. It is worth noting that this action repeats the division by 10 and similar numbers.
Conclusion: it's all about practice
Nothing in learning is easy and effortless. For reliable development of new material requires time and training. Mathematics is no exception.
So that the topic of decimal fractions does not cause difficulties, you need to solve examples with them as much as possible. After all, there was a time when the addition of natural numbers baffled. And now everything is fine.
Therefore, to paraphrase a well-known phrase: decide, decide and solve again. Then tasks with such numbers will be carried out easily and naturally, like another puzzle.
By the way, the puzzles are initially difficult to solve, and then you need to make familiar movements. It is the same in mathematical examples: after going one way several times, then you will no longer think about where to turn.