The concept of a decimal number
In a Formula 1 final decided by hundredths of a second, $1,24,058$ and $1,24,06$ look almost like the same time. $\pi = 3,14159...$ never ends and never repeats. This year you are going to compare decimals with more digits, meet decimals that are not recurring, and understand why between any two decimals there is always a gap with infinitely many numbers inside it.
Ordering several decimals at once
You already know how to compare two decimals digit by digit. With more than two, the method is the same: you make the number of decimal places equal by adding zeros and compare column by column, from left to right.
Order the following decimal numbers from smallest to largest: $$0,52 \quad | \quad 0,509 \quad | \quad 0,6 \quad | \quad 0,521 \quad | \quad 0,5$$
Order the following decimal numbers from smallest to largest:
$$0,52 \quad | \quad 0,509 \quad | \quad 0,6 \quad | \quad 0,521 \quad | \quad 0,5$$In a Formula 1 championship, the qualifying lap times of four drivers around the circuit were as follows: Driver A: $1:24,058\text{ s}$ Driver B: $1:24,5\text{ s}$…
In a Formula 1 championship, the qualifying lap times of four drivers around the circuit were as follows:
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Driver A: $1:24,058\text{ s}$
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Driver B: $1:24,5\text{ s}$
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Driver C: $1:24,06\text{ s}$
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Driver D: $1:24,0578\text{ s}$
Order the drivers from fastest to slowest using the symbols $>$ or $<$ as appropriate.
Order the following decimal numbers from least to greatest, justifying your answer by expanding at least four of their decimal digits: $A = 2,3\hat{5}$…
Order the following decimal numbers from least to greatest, justifying your answer by expanding at least four of their decimal digits:
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$A = 2,3\hat{5}$
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$B = 2,\widehat{35}$
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$C = 2,35$
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$D = 2,35\hat{3}$
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$E = 2,355$
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$F = 2,353$
Once ordered, find a terminating decimal number and a pure recurring decimal number that lie between the two middle values of your list.
Decimals that never end and never repeat: the irrational numbers
$\sqrt{2} = 1{,}4142135...$ However many digits you calculate, you never find a repeating part. This kind of number is neither exact nor recurring: it is irrational.
Classify the following decimal numbers in the attached table (Terminating, Pure recurring, Mixed recurring or Irrational): $3,141592...$ (Number $\pi$) $2,15$ $7,444...$…
Classify the following decimal numbers in the attached table (Terminating, Pure recurring, Mixed recurring or Irrational):
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$3,141592...$ (Number $\pi$)
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$2,15$
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$7,444...$
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$0,1232323...$
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$\sqrt{2}$
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$5,0616161...$
| Terminating | Pure recurring | Mixed recurring | Irrational |
|---|---|---|---|
Represent the following numbers approximately on the number line and determine exactly which of them is the greatest: $X = 1,4\hat{6}$ $Y = \sqrt{2}$ (use the…
Represent the following numbers approximately on the number line and determine exactly which of them is the greatest:
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$X = 1,4\hat{6}$
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$Y = \sqrt{2}$ (use the approximation $1,4142...$)
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$Z = \frac{11}{7}$
Since $\sqrt{2}$ is an irrational number, it has no generating fraction. Therefore:
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For $X$, calculate its generating fraction and compare it with $Z$ (for example, by reducing them to a common denominator).
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For $Y$, use its decimal approximation and compare it with the other two.
Justify the position of each number on the line by doing the necessary divisions with at least three decimal places. Use the line in the figure, where each small mark is worth $0,01$.
You can do it on paper and send a photo with your answer.
The number line is "dense": there is always another decimal in between
Between $1{,}54$ and $1{,}55$ it seems there is no room for anything else. But look one digit further: $1{,}541, 1{,}542, 1{,}543...$ There are infinitely many decimals between two decimals, however close together they are.
Three siblings measure their height. Lucía is $1,54$ metres tall and Marcos is $1,55$ metres tall. Find a decimal number that lies between Lucía's height and Marcos's…
Three siblings measure their height. Lucía is $1,54$ metres tall and Marcos is $1,55$ metres tall. Find a decimal number that lies between Lucía's height and Marcos's height.
Insert three rational numbers (in fraction form) between the decimal numbers $0,\hat{7}$ and $0,7\hat{8}$. To solve it, follow these steps: 1. Find the generating…
Insert three rational numbers (in fraction form) between the decimal numbers $0,\hat{7}$ and $0,7\hat{8}$.
To solve it, follow these steps:
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Find the generating fractions of both numbers.
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Convert these fractions to a common denominator.
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Find three fractions that lie between them and check that their decimal expressions satisfy the insertion condition.
Back to the start: If you make the decimal places equal, $1{,}24,06 = 1{,}24,060$; since the hundredths digit of $1{,}24,058$ is smaller ($5 < 6$), that driver went two thousandths faster and won the position.