Sets of numbers

Unit 2 Sets of numbers

Key unit competence: Use the properties of sets of numbers in solving real life problems.

Key ideas

• Natural numbers $\mathbb{N}=\{1,2,3,\dots \}$ are for counting. Whole numbers include 0. Integers $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$.
• Even numbers are multiples of 2; odd numbers leave remainder 1 when divided by 2; a prime has exactly two factors: 1 and itself (2 is the only even prime; 1 is not prime).
• Adding integers: same signs, add and keep the sign; different signs, subtract and keep the sign of the bigger number. Subtracting is adding the opposite: $a-(-b)=a+b$.
• Multiplying or dividing: same signs give a positive result; different signs give a negative result.
• Decimal numbers have a finite number of decimal places; recurring decimals repeat forever (e.g. $\frac{1}{3}=0.333\dots$).
• Rational numbers $\mathbb{Q}$ can be written $\frac{a}{b}$ with $a,b\in \mathbb{Z}$, $b\ne 0$. Terminating and recurring decimals are rational.
• Irrational numbers (like $\sqrt{2}$, $\pi$) have non-terminating, non-repeating decimals. Rationals and irrationals together form the real numbers $\mathbb{R}$, with $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$.
• Addition and multiplication are commutative and associative; multiplication distributes over addition: $a(b+c)=ab+ac$. Subtraction and division are not commutative. 0 is the identity for addition, 1 for multiplication.

Formulas to know

Common misconceptions

• "$-5$ is bigger than $-2$ because 5 is bigger than 2." No: on the number line $-5$ is to the left of $-2$, so $-5<-2$.
• "$0.333\dots$ is irrational because it never ends." No: it repeats, so it is rational ($\frac{1}{3}$).
• "1 is a prime number." No: a prime has exactly two different factors; 1 has only one.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. From the natural numbers up to 20, list the prime numbers.

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   2, 3, 5, 7, 11, 13, 17, 19.

   Common mistake: Including 1 or 9 (9 = 3 × 3).

2. Show with an example that division is not associative.

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   $(24÷6)÷2=4÷2=2$ but $24÷(6÷2)=24÷3=8$. Since $2\ne 8$, division is not associative.

   Common mistake: Using numbers where both sides happen to agree, like dividing by 1.

3. Work out $15.81-9.36$.

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   $6.45$.

   Common mistake: Not lining up the decimal points.

4. A trader had a debt of 15 000 FRW, then earned 22 500 FRW. Write this as an operation on integers and give her balance.

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   $-15000+22500=+7500$ FRW.

   Common mistake: Ignoring the sign of the debt.

5. Change $\frac{5}{8}$ into a decimal and a percentage.

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   $5÷8=0.625=62.5\%$.

   Common mistake: Dividing 8 by 5 instead of 5 by 8.

6. Work out $\left(\frac{3}{4}-\frac{2}{3}\right)÷\frac{1}{6}$.

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   $\frac{9-8}{12}=\frac{1}{12}$, then $\frac{1}{12}\times 6=\frac{1}{2}$.

   Common mistake: Dividing before working out the bracket.

7. Convert $0.2\overline{7}=0.2777\dots$ to a fraction.

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   $x=0.2777\dots$; $10x=2.777\dots$; $100x=27.777\dots$; $90x=25$, $x=\frac{25}{90}=\frac{5}{18}$.

   Common mistake: Multiplying by 100 only, which does not cancel the non-repeating digit.

8. Classify each number as natural, integer, rational or irrational (give all that apply): $-7$, $\frac{3}{4}$, $\sqrt{16}$, $\pi$, $0$.

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   $-7$: integer, rational. $\frac{3}{4}$: rational. $\sqrt{16}=4$: natural, integer, rational. $\pi$: irrational. $0$: integer (whole), rational. All are real.

   Common mistake: Thinking $\sqrt{16}$ is irrational just because it has a root sign.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Which list contains only prime numbers?	a. 0.6
   2. Write $0.363636\dots$ as a fraction in its simplest form.	b. $3\frac{2}{5}$
   3. Write $\frac{3}{5}$ as a decimal.	c. $\frac{4}{11}$
   4. Write $\frac{17}{5}$ as a mixed number.	d. 2, 3, 5, 7

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   1 → d (2, 3, 5, 7) · 2 → c ($\frac{4}{11}$) · 3 → a (0.6) · 4 → b ($3\frac{2}{5}$)

   • 1 is not prime, 4, 6, 8 and 9 have more than two factors.
   • Step 1. Let $x=0.3636\dots$; then $100x=36.3636\dots$.
     Step 2. Subtract: $99x=36$, so $x=\frac{36}{99}=\frac{4}{11}$.
   • $3÷5=0.6$.
   • Step 1. $17=3\times 5+2$.
     Step 2. So $\frac{17}{5}=3\frac{2}{5}$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   1683  ·  $7^{\circ }$C  ·  $-15^{\circ }$C  ·  2
   1. Work out $\frac{3}{4}÷\frac{3}{8}$. →
   2. The temperature at night was $-4^{\circ }$C and it rose by $11^{\circ }$C during the day. What was the day temperature? →
   3. Use the distributive property to work out $17\times 99$ mentally. →
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   1: 2 · 2: $7^{\circ }$C · 3: 1683

   • Multiply by the reciprocal: $\frac{3}{4}\times \frac{8}{3}=\frac{24}{12}=2$.
   • $-4+11=7$.
   • $17\times (100-1)=1700-17=1683$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Work out $2.5\times 0.4+1.2÷0.3$.	a. $\frac{1}{2}$
   2. Work out $\left(\frac{2}{3}+\frac{1}{4}\right)÷\frac{11}{6}$.	b. $\sqrt{2}$
   3. Which number is irrational?	c. 5
   4. Work out $(-12)÷(+3)$.	d. $-4$

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   1 → c (5) · 2 → a ($\frac{1}{2}$) · 3 → b ($\sqrt{2}$) · 4 → d ($-4$)

   • Step 1. $2.5\times 0.4=1$ and $1.2÷0.3=4$.
     Step 2. $1+4=5$.
   • Step 1. $\frac{2}{3}+\frac{1}{4}=\frac{8}{12}+\frac{3}{12}=\frac{11}{12}$.
     Step 2. $\frac{11}{12}\times \frac{6}{11}=\frac{6}{12}=\frac{1}{2}$.
   • Step 1. $\sqrt{2}=1.41421\dots$ never ends or repeats.
     Step 2. $\sqrt{9}=3$ is rational.
   • Step 1. Different signs give a negative result: $12÷3=4$.
     Step 2. So $-4$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $\frac{3}{2}$  ·  $\frac{17}{30}$  ·  180  ·  $\frac{7}{15}$
   1. Work out $\frac{2}{3}\times \frac{9}{4}$. →
   2. Evaluate $\frac{2}{5}+\frac{1}{3}-\frac{1}{6}$. →
   3. Find the LCM of 12, 18 and 30. →
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   1: $\frac{3}{2}$ · 2: $\frac{17}{30}$ · 3: 180

   • $\frac{2\times 9}{3\times 4}=\frac{18}{12}=\frac{3}{2}$.
   • LCD 30: $\frac{12}{30}+\frac{10}{30}-\frac{5}{30}=\frac{17}{30}$.
   • Step 1. $12=2^2\times 3$, $18=2\times 3^2$, $30=2\times 3\times 5$.
     Step 2. Take the highest powers: $2^2\times 3^2\times 5=180$.

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