Secondary 3 Real Numbers and Number Sets Worksheet

14 of the 15 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 14 of the 15 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Absolute Value Notation — Secondary 1, 2 and 3

   The absolute value of a number is its distance from zero on the number line, so it is never negative.

   1. $|-9|$
   2. $\left|4.5\right|$
   3. $|-\frac{3}{4}|$
   4. Two different numbers both have an absolute value of $12$. Which two?
   5. One January morning it is $-27{}^{\circ }\text{C}$ in Chibougamau and $-8{}^{\circ }\text{C}$ in Montréal. Which reading is further from $0{}^{\circ }\text{C}$, and by how much?
2. Q2Arithmetic

   Arithmetic is the branch of mathematics that studies numbers and the operations carried out on them.

   1. Name the result of each of the four basic operations (the words, not any numbers).
   2. In the fraction $\frac{45}{6}$, name the number written above the line and the one written below it.
   3. Which operation does each of these phrases signal: “increased by”, “shared equally among”, “three times as many”, “how much more than”?
   4. Compute $7+0$, $7\times 1$ and $7\times 0$. Which number leaves any addition unchanged, which leaves any multiplication unchanged, and which turns every product into zero?
3. Q3Ascending and Descending Order
   1. Write in ascending order: $-3.4$, $\frac{7}{2}$, $-\frac{18}{5}$, $0$, $3.25$, $-3$.
   2. Write in descending order: $\sqrt{16}$, $-\frac{9}{4}$, $2.5$, $-2.3$, $\frac{15}{4}$.
   3. Five snow depths measured on the same day are $42\text{cm}$, $\frac{3}{4}$ of a metre, $0.38\text{m}$, $59\text{cm}$ and $\frac{1}{2}$ of a metre. Put them in ascending order, all expressed in centimetres.
4. Q4Integers (\mathbb{Z})

   The integers are the natural numbers together with their opposites: $\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$.

   1. Which of these belong to $\mathbb{Z}$: $-7$, $\frac{8}{2}$, $0$, $-\frac{5}{3}$, $\sqrt{49}$, $4.0$, $-2.5$?
   2. Write the opposite of $-13$, and the opposite of $0$.
   3. A downtown parking garage numbers its levels with integers: level $+3$ is three floors above the ground and level $-3$ is three floors below. A delivery van starts on level $-4$ and climbs $7$ levels. Which level does it reach?
   4. How many integers lie strictly between $-5$ and $3$?
5. Q5Irrational Numbers

   An irrational number cannot be written as a quotient of two integers. In decimal form, its expansion never ends and never falls into a repeating block.

   1. Which of these are irrational: $\sqrt{25}$, $\sqrt{10}$, $\pi$, $0.333\dots$ (the $3$ repeating forever), $\frac{22}{7}$, $-\sqrt{2}$, $1.010010001\dots$ (one extra zero in each group)?
   2. $\sqrt{25}$ and $\sqrt{10}$ end up in different groups. Explain, in one sentence, what separates them.
   3. Give $\sqrt{10}$ to two decimal places, and explain why the value you wrote is not equal to $\sqrt{10}$.
6. Q6Natural Numbers (\mathbb{N})

   The natural numbers are the counting numbers together with zero: $\mathbb{N}=\{0,1,2,3,\dots \}$.

   1. Which of $0$, $-6$, $\frac{12}{4}$, $\frac{7}{2}$, $\sqrt{81}$, $15$, $2.0$ belong to $\mathbb{N}$?
   2. Which of these quantities must be a natural number, and which need not be: the number of buses booked for a school trip; the temperature outside in ${}^{\circ }$C; the number of goals scored in a hockey game; the mass of a backpack in kilograms; a page number in a book?
   3. What is the smallest natural number? Explain why there is no largest one.
7. Q7Number Sets –

   The number sets studied in Secondary 3 sit inside one another: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset ℝ$. The irrational numbers are the real numbers that are not rational.

   1. Say in words what the chain $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset ℝ$ claims.
   2. For each number, name the smallest of the four sets that contains it: $-8$, $\frac{3}{5}$, $\sqrt{7}$, $12$, $-4.25$, $\pi$, $\sqrt{36}$.
   3. True or false, with a reason: every integer is a rational number.
   4. Can a number be both rational and irrational? Explain.
8. Q8Ordering Irrational Numbers
   1. Without a calculator, place each of $\sqrt{3}$, $\sqrt{18}$, $\sqrt{27}$ and $\sqrt{50}$ between two consecutive whole numbers.
   2. Use a) to write those four numbers in ascending order.
   3. Write $\pi$, $\sqrt{8}$, $\sqrt{11}$ and $3.2$ in ascending order, and state the comparison you used for each decision.
9. Q9Ordering Real Numbers
   1. Write in ascending order: $-\frac{7}{2}$, $\sqrt{5}$, $-3.45$, $\frac{9}{4}$, $0$, $-\sqrt{9}$, $2.3$.
   2. Which of the seven numbers are irrational?
   3. Explain why comparing $\sqrt{5}$ with $\frac{9}{4}$ takes more care than comparing $-3.45$ with $0$, and show the comparison that settles it.
10. Q10Rational Numbers (\mathbb{Q})

    A rational number is one that can be written as $\frac{a}{b}$ with $a$ and $b$ integers and $b\ne 0$. In decimal form, its expansion either stops or eventually repeats.

    1. Write each of these as a quotient of two integers: $0.75$, $-6$, $2\frac{3}{8}$, $0.444\dots$, $12\%$.
    2. Write $\frac{5}{8}$ and $\frac{4}{11}$ in decimal form, and say which one stops and which one repeats.
    3. Is $\frac{0}{7}$ a rational number? Is $\frac{7}{0}$? Explain both answers.
11. Q11Real Numbers (\mathbb{R})

    The real numbers are the rational numbers together with the irrational numbers — every number that can be placed on the number line.

    1. Sort into rational and irrational: $\sqrt{64}$, $-\frac{11}{6}$, $\sqrt{6}$, $0.818181\dots$, $3\pi$, $-5.2$.
    2. Give one example each of a real number that is a natural number, an integer that is not natural, a rational number that is not an integer, and an irrational number.
    3. Give one rational number and one irrational number lying strictly between $2.7$ and $2.8$, and say how you know each is what you claim.
12. Q12Representing Number Sets –

    A set of numbers can be given in words, drawn on a number line, or written in interval notation. A square bracket turned towards the interval includes the endpoint; a reversed square bracket excludes it. Infinity is always excluded.

    1. Write in interval notation: all real numbers greater than or equal to $-2$ and less than $6$.
    2. Write in interval notation: all real numbers less than $4$.
    3. Describe in words the set $]-\infty ,-1]$.
    4. Which of $-2$, $0$, $5.9$, $6$ belong to $[-2,6[$?
    5. Write $]3,+\infty [$ as an inequality.
13. Q13The Common Denominator — Secondary 1, 2 and 3
    1. Write $\frac{5}{6}$, $\frac{7}{9}$ and $\frac{11}{12}$ over their least common denominator, then put the three in ascending order.
    2. Compute $\frac{5}{6}+\frac{7}{9}-\frac{11}{12}$, giving the answer in lowest terms.
    3. A canoe route is shared by three paddlers: Léa covers $\frac{3}{8}$ of it and Noah covers $\frac{1}{3}$. What fraction does Zoé, who covers the rest, paddle?
    4. In one sentence, say why fractions need a common denominator to be added but not to be multiplied.
14. Q14Writing Numbers

    This question is built around a diagram or a table of values. Open it in the PDF.

15. Q15Synthesis — drawing on several sheets in this topic

    A weather station's log holds five real numbers: $$A=-\frac{17}{4},B=\sqrt{20},C=-4.2,D=0.666\dots ,E=\frac{9}{2}.$$

    1. Say which of the five are rational and which are irrational, with a reason for each irrational one.
    2. Write all five in ascending order.
    3. Which of the five is furthest from zero? Use absolute value to justify.
    4. Write, in interval notation, the shortest interval whose two endpoints are values from the list and which contains all five.

The 12 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The chain, and the question it lets you answer

The four sets sit inside one another:

Each set keeps everything the one before it had and adds the numbers that operation could not produce. Q7 asks you to say that chain in words and then to name, for each of several numbers, the smallest of the four sets containing it — which is the only version of the question with a single right answer, since 12 is in all four.

Rational or irrational: the three signals, and the one argument

A number is rational exactly when it can be written as a quotient of two integers. In decimal form that shows up as an expansion that either stops or eventually falls into a repeating block. Q5 tests the boundary carefully — the square root of 25 and the square root of 10 sit in different groups, and it asks you to say in one sentence what separates them.

The challenge questions push on the arithmetic of these numbers rather than the labels. Multiplying two irrational numbers sometimes lands back in the rationals and sometimes does not — the square root of 5 times itself is 5 — so the claim "irrational times irrational is irrational" has to be repaired rather than defended. Adding a rational to an irrational, on the other hand, is always irrational, and that one has an argument behind it rather than a list of examples.

Ordering numbers you cannot evaluate

Q3 and Q9 mix fractions, decimals and negatives; Q8 mixes roots. The technique changes with the mixture, and choosing it is most of the work.

Absolute value is a distance

Q1 treats it as exactly that: how far the number sits from zero, so it is never negative, and two different numbers can share one absolute value. The temperature part is the useful one — a January reading of −27 °C is further from zero than one of −8 °C, which is a different question from which reading is larger.

The challenge question tests whether the absolute value of a sum equals the sum of the absolute values. It does when the two numbers point the same way and it fails when they point opposite ways, and the reason is visible on a number line: opposite directions cancel some of the distance before you measure it.

Between any two numbers there is another one

Q11 asks for a rational and an irrational number strictly between 2.7 and 2.8, and to say how you know each is what you claim. The challenge questions go further and ask for a method that works for any two different real numbers, plus the reason it always works — an average is the usual answer, and it is worth being able to say why the average of two different numbers must lie strictly between them.

Then the same claim is tested on the integers, where it is false: 4 and 5 have nothing between them. That contrast is the point. It is what "the number line has no gaps" actually means, and it is why the reals need a different kind of description from the sets that came before.

Writing a collection of numbers down

Q12 introduces interval notation, which is the notation every later course uses for domains, ranges and solution sets. In Quebec notation a square bracket turns outward at an endpoint that is excluded, so [−2, 6[ contains −2 and stops just short of 6, and infinity always takes a reversed bracket because it is never reached.

The challenge question puts it to work on a ski hill's snow cannons, where each endpoint is included or excluded for a physical reason you have to state, and then asks for the temperatures at which exactly one of two hills can make snow. Draw both intervals on the same line and read the answer off; reasoning it out symbolically is much harder than it looks.

The vocabulary and the fractions

Two sheets here are consolidation rather than new material, and both repay a fast pass. Q2 names the results of the four operations, the parts of a fraction and the phrases that signal each operation — the words a question uses when it does not use symbols. Q13 builds least common denominators, adds and subtracts, and asks in one sentence why fractions need a common denominator to be added but not to be multiplied. Q14 puts the same number into fractional, decimal, percentage and scientific notation and asks which one communicates best where.

Getting the most out of it

Ask "what kind of number is this?" before comparing anything

A root, a fraction, a decimal and a percentage are four ways of writing numbers, and the technique for comparing them depends on what you have. Naming them first is what stops you reaching for a calculator on a question that was designed to be settled exactly.

Learn the perfect squares to 15, and the cubes to 5

Every trapping question on this sheet is instant if you know that 36 and 49 sit either side of 40, and slow if you do not. It is a small list and it pays for itself immediately.

Draw the number line for anything negative

Ordering negatives is where careful students lose marks, and a line with zero marked on it removes the whole class of error. Absolute value questions are the same picture read differently: distance from zero, direction ignored.

Say which set, and why, in the same sentence

"Irrational, because its decimal expansion never repeats" is an answer; "irrational" on its own is a guess that happens to be right. Getting into the habit now costs nothing, and every justification question later in high school is marked the same way.

Common questions