Ordinary fractions are used to indicate the ratio of a part to a whole. For example, a cake was shared among five children, so each got a fifth of the cake (1/5).
Ordinary fractions are notations of the form a/b, where a and b are any natural numbers. The numerator is the first or top number, and the denominator is the second or bottom number. The denominator indicates the number of parts by which the whole was divided, and the numerator indicates the number of parts taken.
History of common fractions
Fractions are mentioned for the first time in manuscripts of the 8th century, much later - in the 17th century - they will be called "broken numbers". These numbers came to us from Ancient India, then they were used by the Arabs, and by the 12th century they appeared among the Europeans.
Initially, ordinary fractions had the following form: 1/2, 1/3, 1/4, etc. Such fractions, which had a unit in the numerator and denoted fractions of a whole, were called basic. Many centuries laterthe Greeks, and after them the Indians, began to use other fractions, parts of which could consist of any natural numbers.
Classification of common fractions
There are correct and improper fractions. The correct ones are those in which the denominator is greater than the numerator, and the wrong ones are vice versa.
Every fraction is the result of a quotient, so the fractional line can be safely replaced with a division sign. Recording of this type is used when division cannot be carried out completely. Referring to the example at the beginning of the article, let's say that the child gets part of the cake, not the whole treat.
If a number has such a complex notation as 2 3/5 (two integers and three fifths), then it is mixed, since a natural number also has a fractional part. All improper fractions can be freely converted into mixed numbers by dividing the numerator entirely by the denominator (thus, the whole part is allocated), the remainder is written in place of the numerator with a conditional denominator. Let's take the fraction 77/15 as an example. Divide 77 by 15, we get the integer part 5 and the remainder 2. Therefore, we get the mixed number 5 2/15 (five integers and two fifteenths).
You can also perform the reverse operation - all mixed numbers are easily converted into incorrect ones. We multiply the natural number (the integer part) with the denominator and add it with the numerator of the fractional part. Let's do the above with the fraction 5 2/15. We multiply 5 by 15, we get 75. Then we add 2 to the resulting number, we get 77. We leave the denominator the same, and here is the fraction of the desired type - 77/15.
Reducing ordinaryfractions
What does the operation of reducing fractions imply? Dividing the numerator and denominator by one non-zero number, which will be the common divisor. In an example, it looks like this: 5/10 can be reduced by 5. The numerator and denominator are completely divided by the number 5, and the fraction 1/2 is obtained. If it is impossible to reduce a fraction, then it is called irreducible.
For fractions of the form m/n and p/q to be equal, the following equality must hold: mq=np. Accordingly, fractions will not be equal if equality is not satisfied. Fractions are also compared. Of the fractions with equal denominators, the one with the larger numerator is greater. Conversely, among fractions with equal numerators, the one with the larger denominator is smaller. Unfortunately, all fractions cannot be compared in this way. Often, to compare fractions, you need to bring them to the lowest common denominator (LCD).
NOZ
Let's consider this with an example: we need to compare the fractions 1/3 and 5/12. We work with denominators, the least common multiple (LCM) for the numbers 3 and 12 - 12. Next, let's turn to the numerators. We divide the LCM by the first denominator, we get the number 4 (this is an additional factor). Then we multiply the number 4 by the numerator of the first fraction, so a new fraction 4/12 appeared. Further, guided by simple basic rules, we can easily compare fractions: 4/12 < 5/12, which means 1/3 < 5/12.
Remember: when the numerator is zero, then the whole fraction is zero. But the denominator can never be equal to zero, since you cannot divide by zero. Whenthe denominator is equal to one, then the value of the whole fraction is equal to the numerator. It turns out that any number is freely represented as a numerator and denominator of unity: 5/1, 4/1, and so on.
Arithmetic operations with fractions
Comparison of fractions was discussed above. Let's turn to getting the sum, difference, product and partial fractions:
Addition or subtraction is performed only after reduction of fractions to NOZ. After that, the numerators are added or subtracted and written with the denominator unchanged: 5/7 + 1/7=6/7, 5/7 - 1/7=4/7
- The multiplication of fractions is somewhat different: they work separately with numerators, and then with denominators: 5/71/7=(51) / (77)=5/49.
- To divide fractions, you need to multiply the first by the reciprocal of the second (reciprocals are 5/7 and 7/5). Thus: 5/7: 1/7=5/77/1=35/7=5.
You need to know that when working with mixed numbers, operations are carried out separately with integer parts and separately with fractional ones: 5 5/7 + 3 1/7=8 6/7 (eight integers and six sevenths). In this case, we added 5 and 3, then 5/7 with 1/7. For multiplication or division, you should translate mixed numbers and work with improper fractions.
Most likely, after reading this article, you have learned everything about ordinary fractions, from the history of their occurrence to arithmetic operations. We hope that all your questions have been settled.
