1. PLACE VALUE, ORDERING AND ROUNDING

2. MULTIPLYING AND DIVIDING BY 10, 100 AND 1000 Complete the following: 3.4 × 10 =34 64.34 ÷ = 0.6434 100 × 45.8 = 45 8001000 43.7 × = 4370 100 92.1 ÷ 10 =9.21 73.8 ÷ = 7.38 10 ÷ 1000 = 8.318310 0.64 × = 640 1000 0.021 × 100 =2.1 250 ÷ = 2.5 100

3. MULTIPLYING BY 0.1 AND 0.01 What is 4 × 0.1? We can think of this as 4 lots of 0.1 or 0.1 + 0.1 + 0.1 + 0.1. We can also think of this as 4 ×. 1 10 4 × is equivalent to 4 ÷ 10. 1 10 Therefore: 4 × 0.1 = 0.4 Multiplying by 0.1Dividing by 10 is the same as

4. DIVIDING BY 0.1 AND 0.01 What is 7 ÷ 0.1? We can think of this as “How many 0.1s (tenths) are there in 7?”. There are ten 0.1s (tenths) in each whole one. So, in 7 there are 7 × 10 tenths. Therefore: 7 ÷ 0.1 = 70 Dividing by 0.1Multiplying by 10 is the same as

5. POWERS OF TEN Our decimal number system is based on powers of ten. We can write powers of ten using index notation. 10 = 10 1 100 = 10 × 10 = 10 2 1000 = 10 × 10 × 10 = 10 3 10 000 = 10 × 10 × 10 × 10 = 10 4 100 000 = 10 × 10 × 10 × 10 × 10 = 10 5 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 10 6 10 000 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10 7 …

6. NEGATIVE POWERS OF TEN Any number raised to the power of 0 is 1, so 1 = 10 0 We use negative powers of ten to give us decimals. 0.01 = = = 10 −2 1 10 2 1 100 0.001 = = = 10 −3 1 10 3 1 1000 0.0001 = = = 10 −4 1 10000 1 10 4 0.00001 = = = 10 −5 1 100000 1 10 5 0.000001 = = = 10 −6 1 1000000 1 10 6 0.1 = = =10 −1 1 10 1 10 1

7. STANDARD FORM We can write very large numbers using standard form. For example, the average distance from the earth to the sun is about 150 000 000 km. We can write this number as 1.5 × 10 8 km. To write a number in standard form we write it as a number between 1 and 10 multiplied by a power of ten. A number between 1 and 10 A power of ten

8. These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 10 10 = 50 000 000 000 7.1 × 10 6 = 7 100 000 4.208 × 10 11 = 420 800 000 000 2.168 × 10 7 = 21 680 000 6.7645 × 10 3 = 6764.5 STANDARD FORM – WRITING LARGE NUMBERS

9. How can we write these numbers in standard form? 0.0006 = 6 × 10 −4 0.00000072 = 7.2 × 10 −7 0.0000502 = 5.02 × 10 −5 0.0000000329 = 3.29 × 10 −8 0.001008 = 1.008 × 10 −3 STANDARD FORM – WRITING SMALL NUMBERS

10. ROUNDING WHOLE NUMBERS Complete this table: 37521 274503 7630918 9875 to the nearest 1000 452 to the nearest 100 to the nearest 10 380003750037520 275000274500 763100076309007630920 1000099009880 0500450

11. ROUNDING TO A GIVEN NUMBER OF DECIMAL PLACES Complete this table: 63.4721 87.6564 149.9875 3.54029 0.59999 to the nearest whole number to 1 d.p.to 2 d.p.to 3 d.p. 6363.563.4763.472 8887.787.6687.656 150150.0149.99149.988 43.53.543.540 10.60.600.600

12. ROUNDING TO SIGNIFICANT FIGURES Numbers can also be rounded to a given number of significant figures. The first significant figure of a number is the first digit which is not a zero. For example: 4 890 351 and 0.0007506 This is the first significant figure 0.0007506 This is the first significant figure 4 890 351

13. ROUNDING TO SIGNIFICANT FIGURES For example: 4 890 351 and 0.0007506 This is the first significant figure 0.0007506 This is the first significant figure 4 890 351 The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros. 0.0007506 This is the second significant figure 4 890 351 This is the second significant figure 4 890 351 This is the third significant figure 0.0007506 This is the third significant figure 4 890 351 This is the fourth significant figure 0.0007506 This is the fourth significant figure

14. Complete this table: 6.3528 34.026 0.005708 150.932 to 3 s. f. 0.00007835 to 2 s. f.to 1 s. f. 6.356.46 34.03430 0.005710.00570.006 151150200 0.00007840.0000780.00008 ROUNDING TO SIGNIFICANT FIGURES