Rational numbers
In 3rd ESO you already worked with fractions and decimals. Now it is time to look at them with the precision you will need in Bachillerato: defining exactly what a rational number is, proving why a fraction gives a terminating or recurring decimal, and understanding that between two rational numbers there is always another one, however close together they are.
The set $\mathbb{Q}$
The natural numbers and the integers are particular cases of rational numbers: $5 = \frac{5}{1}$, $-3 = \frac{-3}{1}$. That is why $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$: each set extends the previous one without losing any of its elements.
Classify the following numbers, indicating the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$): $-5$; $\frac{8}{2}$;…
Classify the following numbers, indicating the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$): $-5$; $\frac{8}{2}$; $1,2\widehat{3}$; $\sqrt{7}$.
From fraction to decimal: terminating or recurring?
Dividing the numerator by the denominator always gives a decimal, but its nature depends only on the prime factors of the denominator of the fraction in its simplest form.
Classify the following fractions according to the type of decimal expansion they produce (terminating, pure recurring or mixed recurring) without carrying out the…
Classify the following fractions according to the type of decimal expansion they produce (terminating, pure recurring or mixed recurring) without carrying out the division: $\frac{7}{20}$, $\frac{5}{33}$, $\frac{11}{9}$ and $\frac{3}{14}$.
The generating fraction
Every terminating or recurring decimal can be written as a fraction: that fraction is its generating fraction. The method is based on multiplying by powers of 10 so that the recurring part "cancels out" when we subtract.
Calculate the irreducible generating fraction of the following decimal numbers: $3,45$ and $2,\widehat{7}$.
Calculate the irreducible generating fraction of the following decimal numbers: $3,45$ and $2,\widehat{7}$.
Calculate the irreducible generating fraction of the decimal number $4,1\widehat{36}$.
Calculate the irreducible generating fraction of the decimal number $4,1\widehat{36}$.
Prove mathematically, using the algorithm to obtain the generating fraction, that the pure recurring decimal $0,\widehat{9}$ is exactly equal to the natural number $1$.
Prove mathematically, using the algorithm to obtain the generating fraction, that the pure recurring decimal $0,\widehat{9}$ is exactly equal to the natural number $1$.
Combined operations with rational numbers
Fractions and recurring decimals are operated on just as in 3rd ESO, but now it is best to convert everything to fractions before operating, especially if there are powers, roots or nested brackets: that way the result is exact, with no rounding.
Work out the following fraction operation and simplify the result: $\frac{3}{4} - \frac{1}{2} \cdot \left( \frac{4}{5} + \frac{1}{3} \right)$.
Work out the following fraction operation and simplify the result: $\frac{3}{4} - \frac{1}{2} \cdot \left( \frac{4}{5} + \frac{1}{3} \right)$.
Simplify the following expression using index laws, leaving the result as an irreducible fraction:…
Simplify the following expression using index laws, leaving the result as an irreducible fraction: $\left( \frac{2}{3} \right)^{-2} \cdot \left( \frac{9}{4} \right)^3 \div \left( \frac{3}{2} \right)^5$.
Order and density of $\mathbb{Q}$
Between two different rational numbers, however close together they seem, there are always infinitely many more rational numbers. This property is called density.
Find two distinct rational numbers strictly between $\frac{2}{5}$ and $\frac{3}{5}$.
Find two distinct rational numbers strictly between $\frac{2}{5}$ and $\frac{3}{5}$.