Definition
For any pair of integers a, b with b not zero, the ratio a:b is called a fraction the ratio a:b, it is denoted ab, a is called the
numerator
and b the
denominator.
A fraction is also called a
rational number.
- Note :ab=a:b
- Example :12 = 1:2 = 0,5
Simplifying a fraction
To simplify a fraction we start by
decomposing the numerator and denominator into a product of prime numbers.
When the same number appears in both the numerator and denominator, we can simplify the fraction.
Example : 5632 = 2⋅2⋅2⋅72⋅2⋅2⋅2⋅2 = 74
Irreducible fraction
A fraction is said to be irreducible if its numerator and denominator are prime to each other
To put a fraction into its irreducible form sous sa forme irréductible, we divide the numerator and denominator by their
gcd
.
Equal fractional writing
- When we multiply the numerator and denominator of a fractional writing by the same non-zero number, we obtain a fractional writing that is equal to it.
- When you divide the numerator and denominator of a fractional number by the same non-zero number, you get a fractional number that is equal to it.
Fraction comparison
- Equality of fractions
Two fractions are equal if it is possible to go from one to the other by multiplying or dividing the numerator and denominator by the same number.
- Fractions have the same denominator
Simply compare the numerators.
- Fractions have the same numerators
The largest is the one with the smallest numerator.
- Fractions have different numerators and denominators
We return to the case where the denominators are equal by applying the equality condition of a fraction.
These are the calculation techniques that the
fraction comparator will use in this example to compare the fractions 1911 and 137.
Adding fractions with the same denominator
The sum of two fractions with the same denominator has the same denominator, so its numerator is equal to the sum of the numerators.
Therefore, we have the formula:ak+bk=a+bk
The following example : 13+43 shows how to add two fractions that have the same numeratorr.
Adding fractions with different denominators
We reduce the fractions to the same denominator, to get back to the case of adding fractions with the same denominator.
Subtraction of fractions with the same denominator
The difference of two fractions with the same denominator has the same denominator, its numerator is equal to the difference of the numerators.
Therefore, we have the formula:ak-bk=a-bk
The following example: 43-23 shows how to subtract two fractions that have the same numerator.
Subtracting fractions with different denominators
We reduce the fractions to the same denominator, to get back to the case of subtracting fractions with the same denominator.
Product of fractions
The product of two fractions is equal to the product of the numerators over the product of the denominators.
Example :
34⋅73 = 2112
The following example 34⋅75 : shows how to multiply two fractions.
Division of fractions
Dividing by a fraction is the same as multiplying by the inverse of that fraction, using this rule it is possible to turn a fraction quotient into a fraction product and apply the rules for simplifying a product of fractions.
Example:-8323 = -83⋅32 = -82 = -4