Numbers - where do they come from?  (Page 3/5)

1.1 –8 ; 12 ; 5–11 ; 4 + 0 – $4\left(\frac{1}{2}\right)$ ; $\frac{\text{36}}{9}+\frac{\text{12}}{4}+\frac{\text{22}}{\text{11}}+1$ ; $\sqrt[4]{\text{81}}$ ; $\sqrt{4}+\sqrt{9}$ ; $\sqrt{6}+1$ ; $\sqrt[3]{\text{27}}$

1.2 2.5 – ½ ; $\frac{1}{3}$ ; $1\frac{1}{3}$ ; $\frac{5}{6}-\frac{2}{6}$ ; 0,5 ; 0,05 ; 0,005

1.3 3 ; 3,5 ; 3,14 ; 22  7 ; 355  113 ; 

end of GROUP ASSIGNMENT

CLASS WORK

1 Of course one can write any number in many ways:

• 4 and 8  2 and 1 + 3 and 6 – 2 and $\sqrt{\text{16}}$ and 2 × 2 are the same number!
• 0,5 and $\frac{5}{\text{10}}$ and $\frac{9}{\text{18}}$ and $\frac{\text{50}}{\text{100}}$ and $\sqrt{\frac{1}{4}}$ and $\frac{\sqrt{4}}{\sqrt{\text{16}}}$ are the same.

1.1 Is 1  3 equal to $\mathrm{1,}\dot{3}$ ? What about $\mathrm{1,3}\dot{3}$ ? And 1,33 of 1,333 of 1,3?

1.2 Is $\sqrt{5}$ the same as 2,2? Or 2,24? Or 2,236? Or 2,2361? Or maybe 2,2360? Discuss.

1.3 Is 3 and 3,5 and 3,14 and 22 ÷ 7 and 355 ÷ 113 the same as  ? Make a decision.

2 We can’t always write 3,1415926535897932384626 . . . when we want to use. Why not?

If I have to write down exactly what  is, then I must write  ! The others in question 1.3 are only approximately equal to. But when I have to use  in a calculation to get an answer, then I have to be able to round off properly.

This is π rounded off to different degrees of accuracy :

1 decimal place: 3,1

2 decimal places: 3,14

3 decimal places: 3,142

4 decimal places: 3,1416

5 decimal places: 3,14159

6 decimal places: 3,141593

• You must now ensure that you know how to do rounding off correctly.

3 Simplify and round off the following values, accurate to the number of decimal places given in the brackets.

3.1 3,1  3 (2)

3.2 2 × $\sqrt{2}$ 2)

3.3 5 ×  (2)

3.4 4,5 × $\sqrt{7}$ (0)

3.5 1,000008 + 25  10000 (1)

end of CLASS WORK

How many seconds in a century?

CLASS WORK

1.1 How many hours are there in 17 weeks? 24 × 7 × 17 = 2 856 hours

1.2 How many minutes in a week? 60 × 24 × 7 = 10 080 minutes

1.3 Is it just as easy to calculate how many hours there are in 135 months? Discuss the question in a group and decide which questions have to be answered before the answer can be calculated.

1.4 How many years are there in 173 months? 173  12 = 14,4166 $\dot{6}$ ≈ 14,42 years

• The ≈ sign means “approximately equal to” and is sometimes used to show that the answer has been rounded. It isn’t used a lot, but it is a good habit.

2 Why do we multiply in question 1.1 and 1.2, and divide in question 1.4?

3 How many seconds in a century? It may take a while to get to the answer! How will you know that you can trust your answer?

4.1 There are one thousand metres in a kilometre, so we can say that one metre equals 0,001 kilometres. One metre = 1  1000 kilometres or 1 m = $\frac{1}{\text{1000}}\text{km}$

4.2 There are one thousand millimetres in a metre: 1 mm = $\frac{1}{\text{1000}\times \text{1000}}\text{km}$ = 0,000 001 km

4.3 There are one thousand micrometres in a millimetre: 1 μm = 0,000 000 001 km. (μ is a Greek letter – mu.)

5 Just as we can write very large numbers more conveniently in scientific notation , we also write very small numbers in scientific notation. Below are a few examples of each. Make sure that you can convert ordinary numbers to scientific notation, and vice versa. Calculators also use a sort of scientific notation. They differ, and so you have to make yourself familiar with the way your calculator handles very large and very small numbers.

5.1 1 μm = 0,000 000 001 km So: 1 μm = 1,0 × 10 ^–9 km

• The definition of a light year is the distance that light travels in one year. Because light travels very fast, this is a huge distance. A light year is approximately 9,46 × 10 ¹² km. Write this value as an ordinary number.
• An electron has a mass of approximately 0,000 000 000 000 000 000 000 000 000 91g. What does this number look like in scientific notation?