10^ (-1) = 1/ (10^1) = 1/10 = 0.1
10^ (-2) = 1/ (10^2) = 1/100 = 0.01
10^ (-3) = 1/ (10^3) = 1/1000 = 0.001 etc.

So, the place values to the right of the decimal point are tenths, hundredths, thousandths, and so on. Negative powers of 10 are small numbers, less than 1 and greater than 0.

EXAMPLE

In the decimal 13.97:

1 is in the ‘tens place’.
3 is in the ‘ones place’.
9 is in the ‘tenths place’.
7 is in the ‘hundredths place’.

So:

13.97 = 1 × 10 + 3 × 1 + 9 × 0.1 + 7 × 0.01 = 10 + 3 + 0.9 + 0.07

Binary System

‘Bi’ means two. The binary number system works just like the decimal number system except that the binary number system uses base 2. It uses only two digits, 0 and 1. The binary number system is also referred to as "base 2 system".

The binary system plays an important role in technology and computer science. 0 and 1 can represent off and on, yes and no, etc. Computers, Calculators, Microwave, and other electronic equipments use binary. Binary number system is easy for machines, but, human beings find it very difficult, because it requires so many digits to represent a number in the binary system. For example, the number 213 takes only three digits (2, 1, and 3) to write in the decimal system, yet takes eight digits to write in the binary system (11010101).

All input to the computer and other electronic equipments is converted into binary numbers made up of 0’s and 1’s. But, decimal is what humans are accustomed to, so, the computer and other electronic equipments turn the binary into decimals and display information in a form that humans can easily understand.

The binary system is a positional number system. Each digit in a binary number has a value dependent on its position in the number. A digit's value is the digit multiplied by a power of two according to its position in the number.

For example,

Consider the binary number 10101 (should be read as "one zero one zero one" and not as "ten thousand one hundred one").

In the binary number 10101:

The first 1 on the left is in the sixteen’s place and its value is 1 × 16 = 16.
The first 0 on the left is in the eight’s place and its value is 0 × 8 = 0.
The second 1 is in the four’s place and its value is 1 × 4 = 4.
The second 0 is in the two’s place and its value is 0 × 2 = 0.

The last 1 on the rightmost is in the ones place and its value is 1 × (2 to the zero power) = 1 × (2^0) = 1 × (1) = 1.

To convert a binary number to a decimal, find out the actual value represented by each digit and add them together.

For example, the decimal equivalent of the binary number 10101 (we just discussed) is 16 + 0 + 4 + 0 + 1 = 21.

To convert from decimal to binary:

Step 1: Divide the decimal number by 2.
Step 2: Record the remainder (0 or 1).
Step 3: Repeat Steps 1 and 2 with the quotient until the quotient becomes zero.

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