What does the base signify in a number sustem?

Answer 1

Given that anything raised to the zero-th power is 1, the number 1010 can have many values depending on the base of the number system used in a particular context.

In the decimal system, where the base is 10, the representation 1010 has a value equal to

1(10 raised to 3) + 0(10 raised to 2) + 1(10 raised to 1) + 0(10 raised to 0) which equals 1(1000) + 0(100) + 1(10) + 0(1) = 1000+0+10+0 = 1010 (one thousand ten)

In the binary system used in computers, where the base is 2, the representation 1010 has a value equal to

1(2 raised to 3) + 0(2 raised to 2) + 1(2 raised to 1) + 0(2 raised to 0) which equals 1(8) + 0(4) + 1(2) + 0(1) = 8+0+2+0 = 10 (10 is the equivalent value in decimal).

In the hexadecimal system used in computers, where the base is 16, the representation 1010 has a value equal to

1(16 raised to 3) + 0(16 raised to 2) + 1(16 raised to 1) + 0(16 raised to 0) which equals 1(4096) + 0(256) + 1(16) + 0(1) = 4096+0+16+0 = 5010 (5010 is the equivalent value in decimal).

In the octal system used in computers, where the base is 8, the representation 1010 has a value equal to

1(8 raised to 3) + 0(8 raised to 2) + 1(8 raised to 1) + 0(8 raised to 0) which equals 1(512) + 0(64) + 1(8) + 0(1) = 512+0+8+0 = 520 (520 is the equivalent value in decimal).

In our day to day life we use the decimal system and we understand that the base of the number system is 10 and life runs without a need to know the base since 10(decimal) is implied as the base. But when we have to deal with computers we have to be aware of which base we are using. In the decimal system there are 10 primitive symbols namely, 0,1,2,3,4,5,6,7,8,9 and any number greater than 9 will have to be formed using these (with the associated place value of the digit). Similarly there are 2 primitive symbols in the binary system namely 0,1 and any number greater than 1 will have to be formed using these (with the associated place value of the digit). Similarly there are 16 primitive symbols in the hexadecimal system namely 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F and any number greater than F will have to be formed using these (with the associated place value of the digit). Similarly there are 8 primitive symbols in the octal system namely 0,1,2,3,4,5,6,7 and any number greater than 7 will have to be formed using these (with the associated place value of the digit).