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A R I T H M E T I C

POWERS OF 10

Lesson 1  Section 3

Here is our first example of a problem that not only should not require a calculator, but that we should not even have to write down.  It is an example of skill in arithmetic.

 10. How do we multiply a whole number by a powerof 10? 28 × 10 Add on as many 0's as appear in the power.

 Example 1. 28 × 10 = 280 Add on one 0. 28 × 100 = 2800 Add on two 0's. 28 × 1000 = 28,000 Add on three 0's.

To understand this, look at 28 in the figure, and let us push each digit one place left, so that we have 280.  280 is then ten times 28, because each digit's place value is now ten times more.  (In 28, the '2' tells how many tens, but in 280 it tells how many hundreds.)

Similarly, 2800 = 100 × 28.

Knowing that is an example of skill in arithmetic, which means to be able to do a problem with the most understanding, and therefore the least effort.   Calculating in writing vertically would give the right answer, eventually (we hope).  But it is not skillful.  The student should let go of such written calculations immediately.

To be honest, we can do almost every problem with a calculator.  Yet the student should wonder, What problems should not require a calculator? What problems should an educated person not even have to write down!

Example 2.   608 is 608 ones.  How much are 608 tens?  608 hundreds?  608 thousands?

Answers.  "608 tens" is 608 × 10.  That is the meaning of multiplication (Lesson 8).  608 tens therefore are 6,080.  Simply add on a 0.

608 hundreds = 608 × 100 = 60,800.  Add on two 0's.

608 thousands = 608 × 1000 = 608,000.  Add on three 0's.

Example 3.

a)  50 ones are equal to how many tens?

Answer.  50 ones are simply 50, or 5 tens.

b)   50 tens are how many hundreds?

Answer.  50 tens = 500. (Add on a 0.)  50 tens are 5 hundreds.

c)  50 hundreds are how many thousands?

Answer.  50 hundreds = 5000. (Add on two 0's.)  50 hundreds are 5 thousands.

 11. When a whole number ends in 0's, how do we divide it by a power of 10? 42,000 ÷ 10 Take off as many 0's as appear in the power.

 Examples. 42,000 ÷ 10 = 4,200 Take off one 0. 42,000 ÷ 100 = 420 Take off two 0's. 42,000 ÷ 1000 = 42 Take off three 0's.

Problem.  Reduce this fraction to its lowest terms:

 300 500

Answer.   To reduce a fraction, we must divide both the numerator and denominator by the same number.   Now, if a number ends in 0, then we know that it is divisible by 10; while if a number ends in two 0's, we know that it is divisible by 100.

Both 300 and 500 then are divisible by 100:

 300 500 = 3 5

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