In mathematics, various types of numbers have been studied since their inception. There are a large number of sets and subsets of numbers. Among them are integers, rational, irrational, natural, even, odd, complex and fractional. Today we will analyze information about the last set - fractional numbers.
Definition of fractions
Fractions are numbers consisting of an integer part and fractions of one. Just like integers, there are an infinite number of fractional numbers between two integers. In mathematics, operations with fractions are performed, as with integers and natural numbers. It's quite simple and can be learned in a couple of lessons.
The article presents two types of fractions: ordinary and decimal.
Ordinary fractions
Ordinary fractions are the integer part a and two numbers written with a fractional bar b/c. Common fractions can be extremely handy if the fractional part cannot be represented in rational decimal form. In addition, arithmeticit is more convenient to perform operations through a fractional line. The upper part is called the numerator, the lower part is called the denominator.
Actions with ordinary fractions: examples
The main property of a fraction. When multiplying the numerator and denominator by the same number that is not zero, the result is a number equal to the given one. This property of a fraction helps to bring a denominator for addition (this will be discussed below) or reduce a fraction, making it more convenient for counting. a/b=ac/bc. For example, 36/24=6/4 or 9/13=18/26
Reducing to a common denominator. To bring the denominator of a fraction, you need to represent the denominator in the form of factors, and then multiply by the missing numbers. For example, 7/15 and 12/30; 7/53 and 12/532. We see that the denominators differ by two, so we multiply the numerator and denominator of the first fraction by 2. We get: 14/30 and 12/30.
Compound fractions are ordinary fractions with a highlighted integer part. (A b/c) To represent a compound fraction as a common fraction, you need to multiply the number in front of the fraction by the denominator, and then add it to the numerator: (Ac + b)/c.
Arithmetic operations with fractions
It will not be superfluous to consider known arithmetic operations only when working with fractional numbers.
Addition and subtraction. Adding and subtracting fractions is just as easy as whole numbers, with the exception of one difficulty - the presence of a fractional bar. When adding fractions with the same denominator, it is necessary to add only the numerators of both fractions, the denominators remain withoutchanges. For example: 5/7 + 1/7=(5+1)/7=6/7
If the denominators of two fractions are different numbers, you first need to bring them to a common one (how to do this was discussed above). 1/8 + 3/2=1/222 + 3/2=1/8 + 34/24=1/8 + 12/8=13/8. Subtraction follows exactly the same principle: 8/9 - 2/3=8/9 - 6/9=2/9.
Multiplication and division. Actions with fractions by multiplication occur according to the following principle: numerators and denominators are multiplied separately. In general terms, the multiplication formula looks like this: a/b c/d=ac/bd. In addition, as you multiply, you can reduce the fraction by eliminating the same factors from the numerator and denominator. In another language, the numerator and denominator are divisible by the same number: 4/16=4/44=1/4.
To divide one ordinary fraction by another, you need to change the numerator and denominator of the divisor and perform the multiplication of two fractions, according to the principle discussed earlier: 5/11: 25/11=5/1111/25=511 /1125=1/5
Decimals
Decimals are the more popular and commonly used version of fractional numbers. They are easier to write down in a line or present on a computer. The structure of the decimal fraction is as follows: first the whole number is written, and then, after the decimal point, the fractional part is written. At their core, decimal fractions are compound fractions, but their fractional part is represented by a number divided by a multiple of 10. Hence their name. Operations with decimal fractions are similar to operations with integers, since they are alsowritten in decimal notation. Also, unlike ordinary fractions, decimals can be irrational. This means that they can be infinite. They are written as 7, (3). The following entry is read: seven whole, three tenths in the period.
Basic operations with decimal numbers
Addition and subtraction of decimal fractions. Performing actions with fractions is no more difficult than with whole natural numbers. The rules are exactly the same as those used when adding or subtracting natural numbers. They can also be considered a column in the same way, but if necessary, replace the missing places with zeros. For example: 5, 5697 - 1, 12. In order to perform a column subtraction, you need to equalize the number of numbers after the decimal point: (5, 5697 - 1, 1200). So, the numerical value will not change and it will be possible to count in a column.
Actions with decimal fractions cannot be performed if one of them has an irrational form. To do this, you need to convert both numbers to ordinary fractions, and then use the tricks described earlier.
Multiplication and division. Multiplying decimals is similar to multiplying natural numbers. They can also be multiplied by a column, simply ignoring the comma, and then separated by a comma in the final value the same number of digits as the sum after the decimal point was in two decimal fractions. For example, 1, 52, 23=3, 345. Everything is very simple, and should not cause difficulties if you have already mastered the multiplication of natural numbers.
Division also coincides with the division of naturalnumbers, but with a slight digression. To divide by a decimal number in a column, you must discard the comma in the divisor, and multiply the dividend by the number of digits after the decimal point in the divisor. Then perform division as with natural numbers. With incomplete division, you can add zeros to the dividend on the right, also adding a zero after the decimal point.
Examples of actions with decimal fractions. Decimals are a very handy tool for arithmetic counting. They combine the convenience of natural, whole numbers and the precision of common fractions. In addition, it is quite simple to convert one fraction to another. Operations with fractions are no different from operations with natural numbers.
- Addition: 1, 5 + 2, 7=4, 2
- Subtraction: 3, 1 - 1, 6=1, 5
- Multiplication: 1, 72, 3=3, 91
- Division: 3, 6: 0, 6=6
Also, decimals are suitable for representing percentages. So, 100%=1; 60%=0.6; and vice versa: 0.659=65.9%.
That's all there is to know about fractions. In the article, two types of fractions were considered - ordinary and decimal. Both are pretty easy to calculate, and if you have a complete mastery of natural numbers and operations with them, you can safely start learning fractional ones.
