Fractions and decimals
Your calculator says that $\frac{1}{3} = 0.3333333$ and calmly leaves it at that. But you know that 3 goes on forever. How do you write that without infinite ink? And how do you get the fraction back if all you have is the decimal? You started this in Secondary Year 1; here you review it in depth, and along the way you'll organize in your head all the kinds of numbers you know.
From fraction to decimal
To convert a fraction to a decimal, just divide the numerator by the denominator. What's interesting is what kind of decimal you get:
| Type | Example | What it looks like |
|---|---|---|
| Terminating | $\frac{3}{8} = 0.375$ | It has a finite number of decimal places. |
| Purely repeating | $\frac{5}{11} = 0.454545\ldots = 0.\overline{45}$ | The group of digits that repeats (the period) starts right after the decimal point. |
| Mixed repeating | $\frac{7}{6} = 1.1666\ldots = 1.1\overline{6}$ | Before the period there are digits that don't repeat (the pre-period). |
The period is marked with a bar on top: $0.\overline{45}$. Some books use an arc: $0.\widehat{45}$. 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 there the division gives the same digits again. Look at the "remainder wheel" for $1 \div 7$:
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.
- a) $2.5$
- b) $1.333...$
- c) $4.1252525...$
- d) $0.08$
Look at the following illustration representing the battery level of a cell phone. 1. Write the fraction that represents the charged part of the battery relative to the…
Look at the following illustration representing the battery level of a cell phone.
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Write the fraction that represents the charged part of the battery relative to the total.
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Simplify this fraction to obtain its irreducible fraction.
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If the battery has a total capacity of $4000 \text{ mAh}$, how many mAh of energy are available?
From decimal to fraction: the generating fraction
Every terminating or repeating decimal comes from a fraction, called its generating fraction. Let's 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.
$$3.75 = \frac{375}{100} = \frac{15}{4}$$Repeating. The trick is to multiply by a power of 10 so that, when you subtract, the infinite periods cancel out.
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.
- a) $25\%$
- b) $0.75$
- c) $10\%$
- d) $1.5$
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.
$$ \frac{1.2 - 0,\widehat{6}}{1 - 0.8\widehat{3}} : 3^{-1} $$Percents, fractions, and decimals
A percent is a fraction with denominator 100: 25% means $\frac{25}{100} = \frac{1}{4} = 0.25$. The three forms are the same number.
To find a percent of an amount, multiply by the decimal: 15% of 80 pesos is $80 \times 0.15 = 12$ pesos. To find what percent a part is, divide and multiply by 100: 12 out of 80 is $\frac{12}{80} = 0.15 = 15\,\%$.
In a school of 500 students, a survey was conducted on their favorite sports. The results are as follows: $\frac{1}{2}$ of the students prefer soccer. $\frac{3}{10}$ of…
In a school of 500 students, a survey was conducted on their favorite sports. The results are as follows:
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$\frac{1}{2}$ of the students prefer soccer.
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$\frac{3}{10}$ of the students prefer basketball.
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The rest of the students prefer other sports.
- a) How many students prefer soccer?
- b) How many students prefer basketball?
- c) What fraction of the total number of students prefers "other sports"?
All the numbers: from the naturals to the reals
Every time an operation "couldn't be done," mathematicians expanded the numbers:
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With the natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ you can't do $3 - 5$.
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That's why the integers $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ appear. But with them you can't do $3 \div 5$.
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So we get the rational numbers $\mathbb{Q}$: all the numbers that can be written as a fraction (terminating and repeating decimals).
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And even so, some numbers are missing: the irrational numbers $\mathbb{I}$, decimals with infinitely many digits and no period, like $\sqrt{2} = 1.41421356\ldots$ or $\pi = 3.14159265\ldots$
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Rationals and irrationals 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 the Pythagorean theorem, its diagonal measures $\sqrt{1^2 + 1^2} = \sqrt{2}$. With a compass you carry that diagonal onto the line:
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:
$$ \left( \frac{13}{9} \, , \, 1.4\widehat{5} \right) $$Explain in detail the procedure you followed to find them and prove that they indeed belong to that interval.