Binary numbers--- what does "base-2" mean?

[PeterBushMan]

binary is "base-2" system.

What does "base-2" mean?

 

Is it  -- all numbers are in the form 2^x?

[joigus]

Mentioned as "base-2" on:

https://www.scienceforums.net/search/?q=base-2&quick=1&type=forums_topic

As a topic:

https://www.scienceforums.net/search/?q=base-2&quick=1&type=forums_topic

Mentioned as "binary system" or the like gives similar results.

30 minutes ago, PeterBushMan said:

Is it  -- all numbers are in the form 2^x?

Not exactly. Not all numbers are powers of two, are they?

It means all numbers are expanded as combinations of powers of two with multipliers (digits that are only integers less than two).

That covers all numbers.

[mathematic]

base digits

2......0,1

3......0,1,2

4......0,1,2,3

etc.

[Eise]

'Binary numbers' do not exist, neither do 'decimal numbers'. Numbers exist, but we can use different notations for them.

Take the number 153, in decimal notation. This means:

 (1 x 100) + (5 x 10) + (3 x 1).

With exponents you could write it as:

(1 x 10²) + (5 x 10¹) + (3 x 10⁰).

So the number is expressed as sums of multiples of powers of 10. So the base number is 10.

In binary, 153 is written as 10011001, which means:

(1 x 2⁷) + (0 x 2⁶) + (0 x 2⁵) +(1 x 2⁴) +(1 x 2³) + (0 x 2²) + (0 x 2¹) +(1 x 2⁰) 

Which is in decimal:

(1 x 128) + (0 x 64) + (0 x 32)+ (1 x 16) + (1 x 8 ) + (0 x 4)+ (0 x 2) + (1 x 1) = 153.

Every number can be written in every base.

Edited May 26, 2022 by Eise

[PeterBushMan]

17 hours ago, Eise said:

'Binary numbers' do not exist, neither do 'decimal numbers'. Numbers exist, but we can use different notations for them.

Take the number 153, in decimal notation. This means:

 (1 x 100) + (5 x 10) + (3 x 1).

With exponents you could write it as:

(1 x 10²) + (5 x 10¹) + (3 x 10⁰).

So the number is expressed as sums of multiples of powers of 10. So the base number is 10.

In binary, 153 is written as 10011001, which means:

(1 x 2⁷) + (0 x 2⁶) + (0 x 2⁵) +(1 x 2⁴) +(1 x 2³) + (0 x 2²) + (0 x 2¹) +(1 x 2⁰) 

Which is in decimal:

(1 x 128) + (0 x 64) + (0 x 32)+ (1 x 16) + (1 x 8 ) + (0 x 4)+ (0 x 2) + (1 x 1) = 153.

Every number can be written in every base.

Thanks.