Ordinary fractions are used to indicate the relationship of a part to a whole. For example, the cake was divided between five children, therefore, each got a fifth of the cake (1/5).
Ordinary fractions are records of the form a / b, where a and b are any positive integers. The numerator is the first or upper number, and the denominator is the second or lower. The denominator indicates the number of shares by which the whole was divided, and the numerator indicates the number of shares taken.
History of common fractions
Fractions were first mentioned in manuscripts of the VIII century, much later - in the XVII century - they will be called "broken numbers". These numbers came to us from Ancient India, then the Arabs used them, and by the 12th century they appeared among 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 the whole, were called basic. Many centuries later, the Greeks, and after them, the Indians began to use other fractions, parts of which could consist of any natural numbers.
Classification of fractions
There are fractions right and wrong. The correct ones are those whose denominator is larger than the numerator, while the wrong ones are vice versa.
Every fraction is the result of a particular, therefore, a fractional line can be safely replaced with a division sign. A record of this type is used when division is impossible to complete. Turning to the example at the beginning of the article, we say that the child receives a portion of the cake, and not all the delicacy.
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 irregular fractions can be freely converted into mixed numbers by dividing the numerator entirely by the denominator (thus, the whole part is selected), the remainder is written in the place of the numerator with a conditional denominator. As an example, take the fraction 77/15. Divide 77 by 15, 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 inverse operation - all mixed numbers are easily converted to incorrect ones. The natural number (the integer part) is multiplied with the denominator and added with the numerator of the fractional part. We do the above with a fraction of 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.
We reduce ordinary fractions
What does the fraction reduction operation mean? The division of the numerator and denominator by one non-zero number, which will be a common divisor. For example, it looks like this: 5/10 can be reduced by 5. The numerator and denominator are completely divided by the number 5, and we get the fraction 1/2. If it is impossible to reduce the fraction, then it is called irreducible.
In order for fractions of the form m / n and p / q to be equal, the following equality must be fulfilled: m * q = n * p. Accordingly, fractions will not be equal if equality is not fulfilled. Fractions are also compared. Of fractions with equal denominators, the larger is the one with the greater numerator. And vice versa, of fractions with equal numerators, less is the one with a larger denominator. Unfortunately, all fractions cannot be compared in this way. Often, to compare fractions, you need to bring them to the lowest common denominator (SPD).
NOZ
Consider this as an example: you need to compare the fractions 1/3 and 5/12. We work with denominators, the least common multiple (LCL) for numbers 3 and 12 - 12. Next, we turn to the numerators. Divide the NOC 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 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 in no case can be equal to zero, since it is impossible to divide by zero. When the 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 in the form of the numerator and denominator of the unit: 5/1, 4/1 and so on.
Arithmetic with fractions
Comparison of fractions was considered above. We turn to obtaining the sum, difference, product and partial fractions:
- Addition or subtraction is performed only after the fractions are reduced to SPD. After that, the numerators are added or subtracted and written with the denominator without changes: 5/7 + 1/7 = 6/7, 5/7 - 1/7 = 4/7.
- Fraction multiplication occurs in a slightly different way: they work separately with the numerators, and then with the denominators: 5/7 * 1/7 = (5 * 1) / (7 * 7) = 5/49.
- To divide fractions, you need to multiply the first by a fraction, the inverse of the second (inverse fractions - 5/7 and 7/5). Thus: 5/7: 1/7 = 5/7 * 7/1 = 35/7 = 5.
You need to know that when working with mixed numbers, actions 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, translate mixed numbers and work with irregular fractions.
Most likely, after reading this article, you learned everything about ordinary fractions, from the history of their occurrence to arithmetic operations. We hope that all your questions are settled.