Gre math Review 1 Arithmetic

1.4 Decimals

The decimal number system is based on representing numbers using powers of 10. The place value of each digit corresponds to a power of 10. For example, the digits of the number 7,532.418 have the following place values.

For the digits before the decimal point:
7 is in the thousands place
5 is in the hundreds place
3 is in the tens place
2 is in the ones, or units, place.
And, for the digits after the decimal point:
4 is in the tenths place
1 is in the hundredths place
8 is in the thousandths place.

That is,

the number 7,532.418 can be written as
 
7 times 1,000, +, 5 times 100, +, 3 times 10, +, 2 times 1, +, 4 times the fraction 1 over 10, +, 1 times the fraction 1 over 100, +, 8 times the fraction 1 over 1,000,
or alternatively it can be written as

7 times 10 to the third power, +, 5 times 10 to the second power, +, 3 times 10 to the first power, +, 2 times 10 to the 0 power, +, 4 times 10 to the power negative 1, +, 1 times, 10 to the power negative 2, +, 8 times 10 to the power negative 3.

If there are a finite number of digits to the right of the decimal point, converting a decimal to an equivalent fraction with integers in the numerator and denominator is a straightforward process. Since each place value is a power of 10, every decimal can be converted to an integer divided by a power of 10. Here are three examples:

Example A:  
2.3 = 2 + the fraction 3 over 10, which is equal to 23 over 10
Example B:  
90.17 = 90 + the fraction 17 over 100, which is equal to the fraction with numerator 9,000 + 17 and denominator 100, which is equal to 9,017 over 100
Example C:  
0.612 = 612 over 1,000, which is equal to 153 over 250

Conversely, every fraction with integers in the numerator and denominator can be converted to an equivalent decimal by dividing the numerator by the denominator using long division (which is not in this review). The decimal that results from the long division will either terminate, as in  1 over 4 = 0.25 and 52 over 25 = 2.08, or the decimal will repeat without end, as in  

1 over 9 = 0.111 dot dot dot, 1 over 22 = 0.0454545 dot dot dot, 25 over 12 = 2.08333 dot dot dot.
One way to indicate the repeating part of a decimal that repeats without end is to use a bar over the digits that repeat. Here are four examples of fractions converted to decimals.

Example A:  3 over 8 = 0.375

Example B:  
259 over 40 = 6 +, 19 over 40, which is equal to 6.475
Example C:  
the negative of the fraction 1 over 3 = negative 0.3 with a bar over the digit 3
Example D:  
15 over 14 = 1.0714285 with a bar over the digits 7, 1, 4, 2, 8, and 5

Every fraction with integers in the numerator and denominator is equivalent to a decimal that terminates or repeats. That is, every rational number can be expressed as a terminating or repeating decimal. The converse is also true; that is, every terminating or repeating decimal represents a rational number.

Not all decimals are terminating or repeating; for instance, the decimal that is equivalent to   the positive square root of 2 is   1.41421356237 dot dot dot, and it can be shown that this decimal does not terminate or repeat. Another example is   0.010110111011110111110 dot dot dot, which has groups of consecutive 1’s separated by a 0, where the number of 1’s in each successive group increases by one. Since these two decimals do not terminate or repeat, they are not rational numbers. Such numbers are called irrational numbers.