Numbers - where do they come from?  (Page 2/5)

4 Only terminating and repeating decimal fractions can be written in the form $\frac{a}{b}$ .

4.1 Here are some irrational numbers (check them out on your calculator):

 $\sqrt{2}$ $\sqrt[3]{\text{11}}$ 3,030030003000030…

4.2 These are NOT irrational – explain why not: $\frac{\text{22}}{7}$ ; $\sqrt{\text{25}}$ ; $\sqrt[3]{\text{27}}$

end of ENRICHMENT ASSIGNMENT

Working accurately

CLASS ASSIGNMENT

1 With every question, simplify the numbers, if necessary, and then place each number in its best position on the given number line.

end of CLASS ASSIGNMENT

ENRICHMENT ASSIGNMENT

Inequalities – translating words into maths

1 The number line tells us something very important: If a number lies to the left of another number, it must be the smaller one. A number to the right of another is the bigger.

For example (keep the number line in mind) 4,5 is to the left of 10, so 4,5 must be smaller than 10. Mathematically: 4,5<10.

• –3 is to the left of 5, so –3 is smaller than 5. Mathematically speaking: –3<5
• 6 is to the right of 0, so 6 is bigger than 0 and we write: 6>0 or 0<6, because 0 is smaller than 6.

What about numbers that are equal to each other? Surely 6  3 and $\sqrt{4}$ !

So: 6  3 = $\sqrt{4}$ .

1.1 Use<or>or = between the numbers in the following pairs, without swopping the numbers around:

5,6 and 5,7; 3+9 and 4×3; –1 and –2; 3 and –3 $\sqrt[3]{\text{27}}$ and $\sqrt{\text{15}}$

2 We use the same signs when working with variables (like x and y, etc.). .

For example, if we want to mention all the numbers larger than 3, then we use an x to stand for all those numbers (of course there are infinitely many of them: 3,1 and 3,2 and 3,34 and 6 and 8 and 808 and 1 000 000 etc). So we say: x>3.

• All the numbers smaller than 0: x<0. Like: –1 and –1,5 and –3,004 and –10 etc.
• Numbers larger than or equal to 6: x ≥ 6. Write down five of them.
• All the numbers smaller than or equal to –2: x ≤ –2. Give three examples.

2.1 Use the variable y and write inequalities for the following descriptions:

All the numbers larger than –13,4 All the numbers smaller than or equal to π

3 We extend the idea further:

• All the numbers between 4 and 8: 4<x<8. We also say: x lies between 4 and 8.
• Numbers larger than –3 and smaller than or equal to –0,5: –3<x ≤ –0,5.
• A is larger than or equal to 16 and smaller than or equal to 30: 16 ≤ A ≤ 30.

It works best if you write numbers in the order in which they appear on the number line: the smaller number on the left and the bigger one on the right. Then you simply choose between either<or ≤.

3.1 Now you and a friend must each give three descriptions in words. Then write the mathematical inequalities for one another’s descriptions.

Inequalities – graphical representations

• Once again we use examples 2 and 3 above, but this time we draw diagrams.

  

3.1 Again make your own diagrams.

end of ENRICHMENT ASSIGNMENT

GROUP ASSIGNMENT

1 CALCULATORS ARE NOW FORBIDDEN – DON’T DO ANY SUMS. ESTIMATE THE ANSWERS AS WELL AS YOU CAN AND FILL IN YOUR ESTIMATED ANSWERS. This assignment is the same as before – only you have to draw your own suitable number line for the numbers. First work alone, then the group must decide on the best answer. Fill this answer in on the group’s number line. This group effort is then handed in for marking.