The calculator says that $\frac{1}{3} = 0{,}3333333$ and carries on as if nothing were wrong. But you know that 3 goes on for ever. How do you write that without infinite ink? And how do you get the fraction back if all you have is the decimal? Here you will find out, and along the way you will put all the types of number you know in order in your head.
Before the period there are digits that do not repeat (the non-recurring part).
The period is marked with a bar over the top: $0{,}\overline{45}$. In some books an arc is used: $0{,}\widehat{45}$. They mean the same thing.
Why do the digits repeat?
When you divide by 7, the remainder can only be 0, 1, 2, 3, 4, 5 or 6. If 0 never comes up, sooner or later a remainder repeats, and from then on the division gives the same digits again. Look at the "remainder wheel" for $1 : 7$:
Practice★☆☆Not started
Classify the following decimal numbers into terminating decimals, pure recurring decimals, or mixed recurring decimals. Then, convert the first two (parts a and b) into…
Classify the following decimal numbers into terminating decimals, pure recurring decimals, or mixed recurring decimals. Then, convert the first two (parts a and b) into their corresponding generating fraction.
Look at the following illustration representing the battery level of a mobile phone. 1. Write the fraction that represents the charged part of the battery relative to…
Look at the following illustration representing the battery level of a mobile phone.
Write the fraction that represents the charged part of the battery relative to the total.
Simplify this fraction to obtain its irreducible fraction.
If the battery has a total capacity of $4000 \text{ mAh}$, how many mAh of energy are available?
Every terminating or recurring decimal comes from a fraction, which is called its generating fraction. Let us see how to find it in each case.
Terminating. Put the number without the decimal point in the numerator and a 1 followed by as many zeros as there are decimal places in the denominator. Then simplify.
$$3{,}75 = \frac{375}{100} = \frac{15}{4}$$
Recurring. The trick is to multiply by a power of 10 so that, when you subtract, the infinite periods cancel out.
Practice★☆☆Not started
Calculate the irreducible fraction equivalent to each of the following percentages and decimal numbers. a) $25\%$ b) $0,75$ c) $10\%$ d) $1,5$
Calculate the irreducible fraction equivalent to each of the following percentages and decimal numbers.
Calculate the exact value of the following expression by first converting all decimal numbers into their generating fractions. Do not use decimal approximations during…
Calculate the exact value of the following expression by first converting all decimal numbers into their generating fractions. Do not use decimal approximations during the process.
A percentage is a fraction with denominator 100: 25 % means $\frac{25}{100} = \frac{1}{4} = 0{,}25$. The three forms are the same number.
To calculate a percentage of an amount, multiply by the decimal: 15 % of 80 € is $80 \cdot 0{,}15 = 12$ €. To find what percentage a part is, divide and multiply by 100: 12 out of 80 is $\frac{12}{80} = 0{,}15 = 15\,\%$.
Practice★☆☆Not started
In a school of 500 students, a survey was conducted on their favourite sports. The results are as follows: $\frac{1}{2}$ of the students prefer football. $\frac{3}{10}$…
In a school of 500 students, a survey was conducted on their favourite sports. The results are as follows:
$\frac{1}{2}$ of the students prefer football.
$\frac{3}{10}$ of the students prefer basketball.
The rest of the students prefer other sports.
a) How many students prefer football?
b) How many students prefer basketball?
c) What fraction of the total number of students prefers "other sports"?
All the numbers: from the natural numbers to the real numbers
Every time an operation "could not be done", mathematicians extended the numbers:
With the natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ you cannot do $3 - 5$.
That is why the integers $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ appear. But with them you cannot do $3 : 5$.
This is how the rational numbers $\mathbb{Q}$ arrive: all those that can be written as a fraction (terminating and recurring decimals).
And even so, some numbers are missing: the irrational numbers $\mathbb{I}$, decimals with infinitely many digits and no period, such as $\sqrt{2} = 1{,}41421356\ldots$ or $\pi = 3{,}14159265\ldots$
Rational and irrational numbers together make up the real numbers $\mathbb{R}$: all the points on the number line.
Where is $\sqrt{2}$ on the number line?
Even though its decimals never end, $\sqrt{2}$ has an exact place on the number line. Draw a square with side 1: by Pythagoras' theorem, its diagonal measures $\sqrt{1^2 + 1^2} = \sqrt{2}$. With a compass you transfer that diagonal onto the line:
Practice★★☆Not started
Classify the following numbers into the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimal numbers…
Classify the following numbers into the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimal numbers (terminating or recurring), find their simplified generating fraction.
a) $4,25$
b) $-7$
c) $3,1\widehat{6}$
d) $5,\widehat{27}$
e) $-\frac{18}{3}$
Once you have obtained the fractions, order them from smallest to largest.
Find two rational numbers and two irrational numbers that lie strictly within the interval: $$ \left( \frac{13}{9} \, , \, 1,4\widehat{5} \right) $$ Explain in detail…
Find two rational numbers and two irrational numbers that lie strictly within the interval:
Back to the beginning: The number on the screen is written without infinite ink as $0{,}\overline{3}$ and you recover its fraction by subtracting tails: $10x - x = 3{,}333\ldots - 0{,}333\ldots \Rightarrow 9x = 3 \Rightarrow x = \frac{3}{9} = \frac{1}{3}$.
Summary
Let's practice! 10 medium exercises in this section