In mathematics, various types of numbers are studied from 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.
Fraction Definition
Fractions are numbers consisting of an integer part and fractions of a unit. As well as integers, there is an infinite number of fractional numbers between two integers. In mathematics, actions with fractions are performed, since with integers and natural numbers. It's quite simple and you can learn it in a couple of lessons.
The article presents two types of fractions: ordinary and decimal.
Common fractions
Ordinary fractions represent the integer part a and two numbers written through the fractional line b / c. Ordinary fractions can be extremely convenient if the fractional part cannot be represented in a rational decimal form. In addition, arithmetic operations are more convenient to perform through the fractional bar. The upper part is called the numerator, the lower is the denominator.
Common Fraction Actions: Examples
The main property of the fraction. When multiplying the numerator and denominator by the same number, which is not zero, the result is a number equal to this. This property of the fraction perfectly helps to bring the denominator for addition (this will be described below) or to reduce the fraction, to make it more convenient for counting. a / b = a * c / b * c. For example, 36/24 = 6/4 or 9/13 = 18/26
Reduction to a common denominator. To bring the denominator of a fraction, it is necessary to represent the denominator in the form of factors, and then multiply by the missing numbers. For example, 7/15 and 12/30; 7/5 * 3 and 12/5 * 3 * 2. We see that the denominators are distinguished by two, so we multiply the numerator and denominator of the first fraction by 2. We get: 14/30 and 12/30.
Compound fractions - ordinary fractions with the selected whole part. (A b / c) To represent a composite fraction as an ordinary fraction, it is necessary to multiply the number in front of the fraction by the denominator, and then add it with the numerator: (A * c + b) / c.
Arithmetic with fractions
It will not be superfluous to consider known arithmetic operations only when working with fractional numbers.
Addition and Subtraction. Adding and subtracting ordinary fractions is just as easy as integers, except for one difficulty - the presence of a fractional line. Adding fractions with the same denominator, you need to add only the numerators of both fractions, the denominators remain unchanged. For example: 5/7 + 1/7 = (5 + 1) / 7 = 6/7
If the denominators of the 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 = 1/2 * 2 * 2 + 3/2 = 1/8 + 3 * 4/2 * 4 = 1/8 + 12/8 = 13/8. Subtraction occurs according to exactly the same principle: 8/9 - 2/3 = 8/9 - 6/9 = 2/9.
Multiplication and division. Actions with fractions for multiplication occur according to the following principle: the numerators and denominators are separately multiplied. In general terms, the multiplication formula looks like this: a / b * c / d = a * c / b * d. In addition, as you multiply, you can reduce the fraction, excluding the same factors from the numerator and denominator. In a different language, the numerator and denominator is divided by the same number: 4/16 = 4/4 * 4 = 1/4.
To divide one ordinary fraction into another, you need to change the numerator and denominator of the divisor and multiply the two fractions, according to the principle discussed earlier: 5/11: 25/11 = 5/11 * 11/25 = 5 * 11/11 * 25 = 1/5
Decimal fractions
Decimals are a more popular and frequently used version of fractional numbers. They are easier to write in a line or present on a computer. The structure of the decimal fraction is as follows: first, an integer is written, and then, after the decimal point, the fractional part is written. In essence, decimal fractions are composite ordinary fractions, but their fractional part is represented by a number divided by a multiple of 10. This is where their name came from. Actions with decimal fractions are similar to operations with integers, since they are also written in the decimal number system. Also, unlike common fractions, decimals can be irrational. This means that they can be infinite. They are written like this 7, (3). This record is read: seven point three in a period.
Basic Decimal Actions
Addition and subtraction of decimal fractions. Performing actions with fractions is no more difficult than with integer 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, however, 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 decimal places: (5.5697 - 1.1200). So, the numerical value does not change and can be read in a column.
Decimal fractions cannot be performed if one of them has an irrational look. To do this, you need to translate both numbers into ordinary fractions, and then use the techniques described earlier.
Multiplication and division. Multiplication of decimal fractions is similar to multiplication of natural fractions. They can also be multiplied by a column, simply, without paying attention to the comma, and then separated by a comma in the final value, the same number of digits as the total after the decimal point was in two decimal fractions. For example, 1.5 * 2.23 = 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 division of natural numbers, but with a small digression. To divide by a decimal number in a column, you must discard the comma in the divider, and multiply the dividend by the number of digits after the comma in the divider. Then perform the division as with natural numbers. With incomplete division, you can add zeros to the dividend on the right, also adding zero to the answer after the decimal point.
Examples of actions with decimal fractions. Decimals are a very convenient tool for arithmetic counting. They combine the convenience of natural, integer numbers and the accuracy of ordinary fractions. In addition, it is quite simple to transfer some fractions to others. Actions with fractions do not differ from operations with natural numbers.
- Addition: 1.5 + 2.7 = 4.2
- Subtraction: 3.1 - 1.6 = 1.5
- Multiplication: 1.7 * 2.3 = 3.91
- Division: 3.6: 0.6 = 6
In addition, decimal fractions 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. Two types of fractions were considered in the article - ordinary and decimal. Both are fairly easy to calculate, and if you have completely mastered the natural numbers and actions with them, you can safely proceed to the study of fractional numbers.