By Arthur Wouk
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Extra resources for A course of applied functional analysis
23 Fractions 24 From 23 A. 50000 This answer is reached as follows: 100 500000 = 100 x 500 = 50000 1000 x 1 1 The three noughts of the 1000 on the bottom line cancel out three noughts of the 500000 on the top line leaving us with 500 x 100. We know that to multiply by 100 we add two noughts. Therefore, 500 and another 00 = 50000. Go to 20, work out your answer again. Fractions From 22 A. 1. 2. 3. These were harder examples. 1 we aossed out three noughts in the 5000 on the top, and to equal this on the bottom, we had to cross out two noughts from the 700 and one nought from the 150.
2. 1 10 100 -;- 10000 IS the same as 10 -;- 104 or 10-4 We are now dividing by a base with a negative index, we reverse the sign of the dividing index and multiply by it. Q. What is the answer to: 10 -;- 10- 4 = 10 2 X 10 4 = ? Indices 58 So our answer, 10 6 or 1 000 000 is the same as in the example on frame 57 which we worked out the 'long' way. So remember the rule: When dividing by a base with a negative index, reverse the sign of the dividing index and then multiply by the baseand index instead of dividing.
When we multiply a number by itself however, for instance, 5 x 5, 7 x 7, etc, we can use a shorter method. 5 x 5 is written 52 (spoken, five squared) 7 x 7 is written 7 2 (spoken, seven squared) The figure above and to the right of the number (in this case, the figure 2) is called an index. 52 means 5 x 5 7 2 means 7 x 7 We have written down 5 and 7 the same number of times as the value of the index. In the case of an index of 2 it is called a square. For example, 52 (five squared) 7 2 (seven squared).