Knowledge base · Maths in secondary school

Maths in secondary school

Number sets. The number line

Q: What is the difference between a rational and an irrational number?

A rational number can be written as a fraction p/q, where p and q are integers and q ≠ 0. Its decimal expansion either terminates or repeats. An irrational number, such as √2 or π, cannot be written like that, and its decimal expansion is infinite and non-repeating.

Q: When does a fraction have a terminating decimal expansion?

When, after simplifying, the denominator has no prime factors other than 2 and 5. That is why 7/40 = 0.175 (because 40 = 2 · 2 · 2 · 5), but 7/12 = 0.58(3) (because 12 has a factor 3). You must simplify first: 9/12 = 3/4 = 0.75.

Q: What is the absolute value of a number?

It is the distance of the number from zero on the number line. For a non-negative number it is the number itself, and for a negative one it is its opposite: |5| = 5, |−5| = 5, |0| = 0. An absolute value is never negative.

Q: Why is √2 an irrational number?

If √2 = p/q with the fraction in lowest terms, then p² = 2q², so p would be even. Then p = 2k and 4k² = 2q², that is q² = 2k², so q would be even too. The fraction could be simplified by 2 - yet it was in lowest terms. This contradiction shows that √2 is not rational.

Q: Is 0.(9) the same as 1?

Yes. For x = 0.(9) we have 10x = 9.(9), so 10x − x = 9, that is 9x = 9 and x = 1. It is not an approximation, just two ways of writing the same number.

Natural numbers, integers, rational and irrational numbers, terminating and repeating decimals, turning them into fractions, absolute value and the number line - with step-by-step examples.

In the previous lesson the sets ℕ, ℤ, ℚ and ℝ appeared as examples of sets. Now we will look at them closely. You will see why you cannot always subtract within the natural numbers, how you know that 1/3 has a repeating decimal and 3/8 a terminating one, how to turn 0.(36) into a fraction and how to check without a calculator whether 3√2 is greater than 2√3.

What can you work out within the natural numbers?

The natural numbers are the numbers you count objects with. In Polish schools zero belongs to them too - it is a convention, and many books abroad start the natural numbers at 1:

\mathbb{N} = \{0, 1, 2, 3, 4, \ldots\}

When you add or multiply two natural numbers, the result is always a natural number. We say that addition and multiplication can always be carried out in ℕ. Subtraction and division are different - one example with a result outside ℕ is enough:

\begin{array}{l} 3 - 5 = -2 \notin \mathbb{N} \\ 7 : 2 = 3.5 \notin \mathbb{N} \end{array}

An operation can be carried out in a set when its result for any numbers from that set also belongs to the set. For division we leave out dividing by zero - that cannot be done anywhere.

You can look at the next number sets exactly like that: each one extends the previous one with the numbers that were missing for some operation to always work.

What does the set of integers consist of?

The integers are the natural numbers and their opposites:

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

the negative integers: −1, −2, −3 and so on;
zero - it is neither positive nor negative;
the positive integers: 1, 2, 3 and so on.

The negative integers and the positive integers are two subsets of ℤ. They have no common elements, and together with zero they make up the whole of ℤ. The natural numbers are zero together with the positive integers.

In ℤ you can always add, subtract and multiply. You cannot always divide: 8 : 2 = 4 is an integer, but 7 : 2 = 3.5 is not.

\begin{array}{l} 8 : 2 = 4 \in \mathbb{Z} \\ 7 : 2 = 3.5 \notin \mathbb{Z} \end{array}

Try it yourself

Which of these results belongs to ℕ: 5 − 9, (−3) · (−4) or 20 : 8?

Hint: Work out each result first, and only then check whether it is a natural number.

Show the answerHide the answer

Answer

\begin{array}{l} 5 - 9 = -4 \notin \mathbb{N} \\ (-3) \cdot (-4) = 12 \in \mathbb{N} \\ 20 : 8 = 2{,}5 \notin \mathbb{N} \end{array}

Only (−3) · (−4) = 12. The difference 5 − 9 is negative, and the quotient 20 : 8 = 2.5 is not an integer. Notice that multiplying two negative numbers gave a natural number.

What is a rational number?

A rational number is a number that can be written as a fraction p/q, where p and q are integers and q ≠ 0. The set of rational numbers is denoted ℚ.

In the fraction p/q, p is the numerator and q the denominator. The denominator tells you how many equal parts the whole is split into, and the numerator how many of those parts you take. The denominator cannot be zero, because we never divide by zero.

It is not only “ordinary” fractions that are rational. Every integer is rational too, because you can write it with denominator 1. Decimals and mixed numbers are rational as well:

-3 = \frac{-3}{1}

0.75 = \frac{3}{4}

1\tfrac{1}{2} = \frac{3}{2}

The same number can be written as many different fractions: 3/4 = 6/8 = 75/100. A fraction that cannot be simplified any further is said to be in lowest terms.

\frac{3}{4} = \frac{6}{8} = \frac{75}{100}

In ℚ all four operations can be carried out: addition, subtraction, multiplication and division (except division by zero). The sum, difference, product and quotient of two fractions is again a fraction.

What decimal expansions do rational numbers have?

You get the decimal expansion of a fraction by dividing the numerator by the denominator. Sometimes the division comes to an end - then the expansion terminates:

3 : 8 = 0.375

7 : 20 = 0.35

7 : 40 = 0.175

Sometimes the division never ends, but the digits start repeating. The repeating block is called the period (or repetend) and is written in brackets. The expansion is then infinite and repeating:

0{,}(3) = \tfrac{1}{3}

0{,}8(3) = \tfrac{5}{6}

0{,}(285714) = \tfrac{2}{7}

In 5/6 = 0.8333… the period is 3, and the 8 comes before the period. In 2/7 the period is as long as six digits: 285714.

Why must a period appear?

When you divide by q, the remainders can only be 0, 1, 2, …, q − 1. If a zero remainder never appears, then within at most q steps some remainder must repeat - and from that point the division goes exactly as before, so the digits start repeating. That is why for 2/7 the period cannot be longer than 6 digits.

Every rational number has a decimal expansion that either terminates or is infinite and repeating. And conversely: every such expansion is a rational number.

How can you tell without dividing whether the expansion terminates?

Simplify the fraction and break the denominator into prime factors. The expansion terminates exactly when there are no prime factors other than 2 and 5. That is because 10, 100, 1000 and every other power of ten are made only of 2s and 5s - so only such a denominator can be scaled up to one of them.

\frac{7}{40} = \frac{175}{1000} = 0.175

40 = 2 \cdot 2 \cdot 2 \cdot 5

Always simplify first. 12 has a factor 3, and yet 9/12 terminates: 9/12 = 3/4 = 0.75. The 3 disappeared when you simplified.

\frac{9}{12} = \frac{3}{4} = 0.75

Game: terminating or repeating?

Score: 0

What kind of decimal expansion does this fraction have?

Question 1/10

How do you turn a decimal into a fraction?

A terminating decimal is written as a fraction with denominator 10, 100 or 1000 - as many zeros as there are digits after the point - and then simplified:

0.375 = \frac{375}{1000} = \frac{3}{8}

For a repeating decimal there is a trick with shifting the decimal point. Call the number x and multiply it by 10, 100 or 1000 - so that the repeating block moves by exactly its own length. When you subtract the two equations, the infinite tails are identical and cancel out:

1
$x = 0{,}(27)$ call the number x
2
$100x = 27{,}(27)$ the period has 2 digits, so multiply by 100
3
$100x - x = 27$ subtract - the tails after the point cancel
4
$99x = 27$ tidy up the left-hand side

\frac{27}{99} = \frac{3}{11}

0{,}(27) = \tfrac{3}{11}

What are irrational numbers?

An irrational number is a real number that cannot be written as a fraction p/q with integers p and q. Its decimal expansion is infinite and non-repeating - the digits never settle into a fixed repeating pattern.

\begin{array}{l} \pi \approx 3.14159 \\ \sqrt{2} \approx 1.41421 \\ \frac{\sqrt{3}}{2} \approx 0.86603 \end{array}

π is the ratio of a circle’s circumference to its diameter - the same for every circle. √2 is the number that gives 2 when squared; it is the length of the diagonal of a square with side 1. And √3/2 is, for example, the height of an equilateral triangle with side 1.

How do we know that √2 is irrational?

Suppose that √2 = p/q after all, with the fraction in lowest terms. Squaring gives p² = 2q², so p² is even - and then p is even too (the square of an odd number is odd). Write p = 2k. Then 4k² = 2q², that is q² = 2k², so q is even as well. Since p and q are both even, the fraction p/q can be simplified by 2 - yet it was supposed to be in lowest terms. The assumption leads to a contradiction, so √2 is not a rational number.

In the same way you can prove that √3 is irrational. And since √3 is irrational, so is √3/2: if √3/2 were a fraction, then √3 = 2 · √3/2 would be a fraction too.

Careful - not every root is irrational. √9 = 3 and √(4/9) = 2/3 are rational. The square root of a natural number is rational only when that number is the square of a natural number. And the popular 22/7 is only an approximation of π, not π itself: 22/7 has a repeating expansion.

3{,}(142857) = 3\tfrac{1}{7}

How do the number sets fit inside one another?

The rational and irrational numbers together make up the set of real numbers ℝ. Every natural number is an integer, every integer is rational and every rational number is real. The irrational numbers are the difference ℝ \ ℚ: the real numbers that are not rational.

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Each number stands in the smallest set it belongs to - and therefore in every set that contains it. √9 = 3, so it goes into ℕ.

\begin{array}{l} \sqrt{9} = 3 \in \mathbb{N} \\ -4 \notin \mathbb{N},\ -4 \in \mathbb{Z} \\ -2.5 \notin \mathbb{Z},\ -2.5 \in \mathbb{Q} \\ \sqrt{2} \notin \mathbb{Q},\ \sqrt{2} \in \mathbb{R} \end{array}

What is the absolute value of a number?

The absolute value of a number a is its distance from zero on the number line. We write it |a|. The numbers 5 and −5 lie on opposite sides of zero, but both at a distance of 5:

|5| = 5

|-5| = 5

|0| = 0

|-3.2| = 3.2

In terms of calculation: the absolute value of a non-negative number is the number itself, and of a negative number - its opposite:

|a| = a \quad (a \ge 0)

|a| = -a \quad (a < 0)

The minus in front of a does not mean a negative number here, it means the opposite number. For a = −5 we get −a = −(−5) = 5. That is why an absolute value is never negative.

The distance between two numbers

The distance between a and b on the number line is |a − b|. The order does not matter, because |a − b| = |b − a|. For example, −2 and 3 are 5 units apart:

|-2 - 3| = |-5| = 5 = |3 - (-2)|

Try it yourself

Work out |1 − √2|.

Hint: First work out the sign of the number inside. Is √2 greater than 1?

Show the answerHide the answer

Answer

|1 - \sqrt{2}| = \sqrt{2} - 1

Since √2 > 1 (because 2 > 1), the number 1 − √2 is negative. Its absolute value is the opposite number: −(1 − √2) = √2 − 1.

How does the number line show the set of real numbers?

A number line is a straight line with a chosen point 0, a unit segment and a direction. Every real number corresponds to exactly one point on it, and every point to exactly one real number. A bigger number always lies to the right of a smaller one.

The rational numbers lie very densely on the line: between any two of them there are infinitely many others. For example, halfway between a and b lies their average (a + b)/2, and if a and b are rational, so is the average. Even so, the rational numbers do not fill the line. It still has irrational points - √2, for example. You can mark it exactly: draw a square with side 1 on the segment from 0 to 1 and use a compass to carry its diagonal over onto the line.

Worked examples, step by step

Example 1

Give the decimal expansions of rational numbers

Write the numbers 3/8, 7/20, 5/6 and 2/7 as decimals.

Solution

1
All four fractions are in lowest terms, so you look straight at the denominators. 8 = 2 · 2 · 2 and 20 = 2 · 2 · 5 have no prime factors other than 2 and 5 - these expansions will terminate. Divide the numerator by the denominator:
$\begin{array}{l} 3 : 8 = 0{,}375 \\ 7 : 20 = 0{,}35 \end{array}$
2
6 = 2 · 3 has a factor 3, so 5/6 repeats. When you divide, after the digit 8 the remainder is 2 and from then on it repeats forever - the period is 3:
$0{,}8(3) = \tfrac{5}{6}$
3
The number 7 is a prime other than 2 and 5, so 2/7 repeats too. The successive remainders are 2, 6, 4, 5, 1, 3 and then 2 again - the digits repeat every six places:
$0{,}(285714) = \tfrac{2}{7}$

Answer: 3/8 = 0.375, 7/20 = 0.35, 5/6 = 0.8(3), 2/7 = 0.(285714).

Example 2

Find the fraction with a terminating decimal expansion

Write the numbers 0.45 and 2.125 as fractions in lowest terms.

Solution

1
0.45 has two digits after the point, so you write it in hundredths. Both numerator and denominator are divisible by 5:
$0{,}45 = \frac{45}{100} = \frac{9}{20}$
2
2.125 has three digits after the point, so you write it in thousandths. 2125 = 125 · 17 and 1000 = 125 · 8, so you simplify by 125:
$2{,}125 = \frac{2125}{1000} = \frac{17}{8} = 2\tfrac{1}{8}$

Answer: 0.45 = 9/20 and 2.125 = 17/8.

Example 3

Find the fraction with a repeating decimal expansion

Write the numbers 0.(36) and 1.2(3) as fractions.

Solution

1
Let x = 0.(36). The period has two digits, so you multiply both sides by 100 and get 100x = 36.(36). Subtract x - the tails after the point are the same and vanish:
$100x - x = 36$
2
So 99x = 36. Divide and simplify by 9:
$\frac{36}{99} = \frac{4}{11}$
$0{,}(36) = \tfrac{4}{11}$
3
Now x = 1.2(3). Here there is also a digit 2 before the period. First move the point past it: 10x = 12.(3). Then one more place, by the length of the period: 100x = 123.(3). Subtract these two equations:
$100x - 10x = 123 - 12$
4
That gives 90x = 111. Divide and simplify by 3:
$\frac{111}{90} = \frac{37}{30} = 1\tfrac{7}{30}$
$1{,}2(3) = 1\tfrac{7}{30}$

Answer: 0.(36) = 4/11 and 1.2(3) = 37/30.

Example 4

Compare irrational numbers without a calculator

Compare the numbers a = 3√2 and b = 2√3. Then compare π and √10.

Solution

1
Both a and b are positive. For positive numbers the bigger number has the bigger square, so it is enough to compare the squares - and those are natural numbers:
$\begin{array}{l} (3\sqrt{2})^2 = 9 \cdot 2 = 18 \\ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \end{array}$
2
Since 18 > 12, a > b.
$3\sqrt{2} > 2\sqrt{3}$
3
For π and √10 you again compare squares. You do not know π exactly, but you know that π < 3.15. Then π² < 3.15² - and that you can work out on paper:
$3{,}15^2 = 9{,}9225 < 10$
4
So the square of π is less than 10, that is less than the square of √10. Both numbers are positive, so:
$\pi < \sqrt{10}$

Answer: 3√2 > 2√3 and π < √10.

Where is this useful in everyday life?

Absolute value

A change in temperature. In the morning it was −7 °C, in the afternoon 5 °C. The temperature changed by |5 − (−7)| = 12 degrees - the absolute value tells you by how much, whatever the direction.
Tolerances. On a technical drawing, “50 mm ± 0.2 mm” means that a part’s dimension x is acceptable when |x − 50| ≤ 0.2 - the deviation in either direction does not exceed 0.2 mm.
Floors in a building. From the car park on level −2 to the 5th floor, the lift goes up |5 − (−2)| = 7 levels.

Converting between fractions and decimals

In the kitchen a recipe asks for 3/4 of a cup, but the measuring jug has decimal markings - that is 0.75.
In a shop, “one fifth off” is a discount of 1/5 = 0.2, that is 20%.
Splitting a 100 zł bill between three people gives 33.(3) zł. You cannot split grosze forever, so each of you pays 33.33 zł - that is 99.99 zł, and there is still one grosz left over. The repeating decimal tells you straight away that it cannot be split evenly.

Comparing irrational numbers

Will a 4.5 m plank fit flat on the floor of a 3 m by 3 m square room if you lay it along the diagonal? The diagonal is 3√2 m. Compare the squares: (3√2)² = 18, while 4.5² = 20.25. The diagonal is shorter than the plank - it will not fit.
The A paper sizes are designed so that the longer side is √2 times the shorter one - then a sheet folded in half keeps the same proportions. An A4 sheet is 297 mm by 210 mm. Is 297 : 210 exactly √2? Compare the squares: 297² = 88 209, while 2 · 210² = 88 200. It comes out slightly bigger, so the ratio is just a little more than √2 - because the sizes are rounded to whole millimetres.

A puzzle to finish

Riddle

The number 0.(9) is 0.999… - nines forever. It seems to be a tiny bit less than 1.

Is 0.(9) less than 1 or equal to 1?

Show the solutionHide the solution

Solution

0{,}(9) = 1

It is equal to 1. Let x = 0.(9). Then 10x = 9.(9), and subtracting gives 10x − x = 9, that is 9x = 9 and x = 1. You can check it another way: 1/3 = 0.(3), and 3 · 1/3 = 1, so 3 · 0.(3) = 0.(9) must also equal 1. It is not an approximation - it is two ways of writing the same number.

Remember

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. In ℕ you can always add and multiply, in ℤ also subtract, and in ℚ carry out all four operations (except dividing by zero). A rational number is a fraction p/q with integers p and q, q ≠ 0; its decimal expansion terminates or repeats, and it terminates exactly when the denominator of the fraction in lowest terms has no prime factors other than 2 and 5. Irrational numbers such as π, √2 and √3/2 have infinite non-repeating expansions. |a| is the distance of a from zero on the number line, and |a − b| is the distance between a and b.

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Frequently asked questions

What is the difference between a rational and an irrational number?

A rational number can be written as a fraction p/q, where p and q are integers and q ≠ 0. Its decimal expansion either terminates or repeats. An irrational number, such as √2 or π, cannot be written like that, and its decimal expansion is infinite and non-repeating.

When does a fraction have a terminating decimal expansion?

When, after simplifying, the denominator has no prime factors other than 2 and 5. That is why 7/40 = 0.175 (because 40 = 2 · 2 · 2 · 5), but 7/12 = 0.58(3) (because 12 has a factor 3). You must simplify first: 9/12 = 3/4 = 0.75.

How do you turn a repeating decimal into a fraction?

Call the number x and multiply it by 10, 100 or 1000 - so that the repeating block shifts by its whole length. When you subtract x, the repeating part cancels. For x = 0.(27): 100x = 27.(27), so 99x = 27 and x = 27/99 = 3/11.

What is the absolute value of a number?

It is the distance of the number from zero on the number line. For a non-negative number it is the number itself, and for a negative one it is its opposite: |5| = 5, |−5| = 5, |0| = 0. An absolute value is never negative.

Why is √2 an irrational number?

If √2 = p/q with the fraction in lowest terms, then p² = 2q², so p would be even. Then p = 2k and 4k² = 2q², that is q² = 2k², so q would be even too. The fraction could be simplified by 2 - yet it was in lowest terms. This contradiction shows that √2 is not rational.

Is 0.(9) the same as 1?

Yes. For x = 0.(9) we have 10x = 9.(9), so 10x − x = 9, that is 9x = 9 and x = 1. It is not an approximation, just two ways of writing the same number.

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