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# Binary And Decimal

Hi CLN

in Binary

What is The 1st Number ? The Red Circle OR The Blue Circle ?

And What Is The 4th Number ? The Red Circle OR The Blue Circle ?

also in Decimal

What is The 1st Number ? The Red Circle OR The Blue Circle ?

And What Is The 4th Number ? The Red Circle OR The Blue Circle ?

• ###### 1. Re: Binary And Decimal

Mohamed,

Slight correction: The number is the entire thing. What you are talking about is digits.

Usually all numbers are evaluated left to right, so the first digit of every number is the leftmost one regardless of base system. However, leading zeros don't usually count, because you can have as many of them as you want without affecting the number's value. So it makes sense to only count weighted digits.

• ###### 2. Re: Binary And Decimal

Slight correction: The number is the entire thing. What you are talking about is digits.

i don`t understand

Usually all numbers are evaluated left to right, so the first digit of every number is the leftmost one regardless of base system. However, leading zeros don't usually count, because you can have as many of them as you want without affecting the number's value. So it makes sense to only count weighted digits.

iam really sorry but i still don`t understand you

please draw what you want to say . OR explain on the picture what you want to explain

you mean red Circle OR Blue Circle ???

• ###### 3. Re: Binary And Decimal

OK,

So... The examples of decimal NUMBERS: 10,  5548, 99804.

The examples of decimal DIGITS: 9,   0,  2,   1.

The difference between numbers and digits is that a NUMBER- is some mathematical VALUE. The DIGIT - is a sign that can be used to write the number down. It is like a word and a letter. A word have some meaning. A letter on its own doesn't have a meaning. It is a sign used to write down words. So, you are looking at number 5878 and say that it consists of four digits: 5, 8, 7 and 8 again. First digit of that number is 5, second digit of that number is 8, third digit is 7 and fourth digit is 8. If you write this number like that: 005878, then you have more digits in it, but its value didn't change. In other words 005858 = 5878. So, adding leading zeros doesn't change the number in any way. That's why you only evaluate the digits that have some weight to them.

• ###### 4. Re: Binary And Decimal

So... The examples of decimal NUMBERS: 10,  5548, 99804.

The examples of decimal DIGITS: 9,  0,  2,  1.

The difference between numbers and digits is that a NUMBER- is some mathematical VALUE. The DIGIT - is a sign that can be used to write the number down. It is like a word and a letter. A word have some meaning. A letter on its own doesn't have a meaning A letter on its own doesn't have a meaning. It is a sign used to write down words

Now I Understand You.

Very Good Example

Very Good Example

Very Good Example

Very Good Example

Very Good Example

Very Good Example

but i have another question , you said that 5878 The 1st Number is "5" and The Second is "8"

So, you are looking at number 5878 and say that it consists of four digits: 5, 8, 7 and 8 again. First digit of that number is 5, second digit of that number is 8, third digit is 7 and fourth digit is 8.

Why ?

Why 1st Number Is "5" NOT "8" ???

why you begin from Left NOT Right ???

i don`t understand this point

• ###### 5. Re: Binary And Decimal

Mohammed, first DIGIT is 5, not first NUMBER

The first one is always at the left because it has the highest weight. How do you write  number 5878 by words? Five thousand, eight hundred and seventy eight. So, you start with the "heaviest" number, which is at the left. Or in the language of mathematics, this number consist of four digits with the following weights:

5*10^3 + 8*10^2 + 7*10^1 + 8*10^0

Every number in any base can be represented as a series of digits multiplied by the base raised to the power equal to the digit's position in the number, where position counts from the right and start with zero.

So, here is how number 5878 breaks down: we are using base10, so we multiply every digit by 10 and raise the base (10) to the power of digits position. We've got 4 digits, so 4 positions: 0, 1, 2 and 3. So, take each digit, separately, multiply it by 10 raised to the power of digit's position and add the results. You will end up with your number 5878

• ###### 6. Re: Binary And Decimal

Sergey already gave some nice eplanations, but let's have a look at some examples. First in decimal, because i think, you know decimal much better than binary until now:

let's talk about the number 7315 (it is the complete number in the "decimal" part of your picture above, the red circle marks only the leftmost digit of that number, while the blue circle marks the rightmost digit of that number). How do we determine the value of that number? We know, that the digits have different meanings depending on their position in the number. The righmost digit counts 1, the digit left to it counts 10, the next digit to the left counts 100 (i.e. 10*10) and so on. So the number 7315 is nothing else than

5*1 + 1*10 + 3*100 + 7*1000

(read from right to left, and each digit counts 10 times than it left neighbor, while the rightmost digit counts 1).

Now let's see, if the value changes, if we add additional 0 digits to the left: 007315 is

5*1 + 1*10 + 3*100 + 7*1000 + 0*10000 + 0*100000,

but since 0*10000 + 0* 100000 ist still nothing else than 0, the zeros added to the left do not change the value. Thats, how our decimal number system works. To write all possible values in decimal system, we need 10 different digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9).

Now let's look at a similar sample with the binary number system. Now we only have 2 digits (0 and 1), and every digit in an number (which may consist of several digits) counts twice than its right neighbor while the righmost digit counts 1. So the value of the binary number 0101 (the number in the "binary" section of your picture) can be determined as (again: read from right to left the same way we did for the decimal number):

1*1 + 0*2 + 1*4 + 0*8

The leading 0 digit does not change the value, so we can omit it, if we want. If we determine the value, we see, that this number is nothing else than the decimal number 5. The blue circle in your picture marks the left most digit (which is 0 in your example, and which can be omited, as we saw, that 0 digits added to the left of a number will not change the value) while the red circle marks the rightmost digits (which counts 1). So the binary number system seems to be more complicated at the first look, but it's nearly the same as the decimal system, but it has only 2 digits and the "value" of a digt is not 10 times of the value of its "right neighbor" as in decimal but only 2 times than its "right neighbor".

Now we can have a look at hexadecimal numbers (even if that is not shown in your picture): it is again the same principle as in decimal and binary number systems, but each digit count 16 times more than its right neighbor, and so we need 16 digits to be able to write each number in hexadecimal. So we use the characters a, b, c, d, e and f as additional digits to the 10 decimal digits we already know (a in hey is 11 in decimal, b in hey is 12 in decimal, c in hex is 13 in decimal, d in hey is 14 in decimal and f in hex is 15 in decimal). Let's have  a look to the hex number 13af (again read from right to left to determine its decimal value):

f(hex)*1 + a(hex)*16(dec) + 3*256(dec) + 1*4096(dec) = (in decimal) 15*1 + 10*16 + 3*256 + 1*4096 = 5039

So even the hexamdecimal number system works the same way. Only the base of the number system is different (and the number of different digits we need is identical to the base of the number system).

I hope, this explains a little bit about how the different number systems work. In IT, we usually only need binary (base 2), decimal (base 10) and hexadecimal (base 16) number systems, but it is also possible to construct number systems with other bases, for example with base 5 or base 7.

• ###### 7. Re: Binary And Decimal

My Dear Friend

Mohammed, first DIGIT is 5, not first NUMBER

Yes, You`re Right , i totally understand you

The first one is always at the left because it has the highest weight. How do you write  number 5878 by words? Five thousand, eight hundred and seventy eight. So, you start with the "heaviest" number, which is at the left. Or in the language of mathematics, this number consist of four digits with the following weights:

Good Good, Now I Understand This point

5*10^3 + 8*10^2 + 7*10^1 + 8*10^0

what is that ? i don`t understand this line

Every number in any base can be represented as a series of digits multiplied by the base raised to the power equal to the digit's position in the number, where position counts from the right and start with zero.

No, I don`t understand What You Want To Say in these lines

So, here is how number 5878 breaks down: we are using base10, so we multiply every digit by 10 and raise the base (10) to the power of digits position. We've got 4 digits, so 4 positions: 0, 1, 2 and 3. So, take each digit, separately, multiply it by 10 raised to the power of digit's position and add the results. You will end up with your number 5878

No, I don`t understand anything

please provide any picture or draw when you explain this point to be easier

• ###### 8. Re: Binary And Decimal

(read from left to right, and each digit counts 10 times than it left neighbor, while the leftmost digit counts 1).

this bold word , i think you mean rightmost NOT leftmost

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another typo

(again: read from right to left the same way we did for the decimal number):

i think you want to say read from Left To Right

• ###### 9. Re: Binary And Decimal

Yes, you are right. Thanks for telling me that. I have corrected this one. I hope, that this is helpful to understand the different numbering systems and how to determine the value of numbers in that different number systems.

• ###### 10. Re: Binary And Decimal

Mohamed,

Watch this Youtube video and it will hopefully help you better understand how different counting systems work and what they have in common.

• ###### 11. Re: Binary And Decimal

When We Say the 1st is the blue circle ?

• ###### 12. Re: Binary And Decimal

Mohamed schrieb:

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another typo

(again: read from right to left the same way we did for the decimal number):

i think you want to say read from Left To Right

No, the typo (now corrected) was before ...

I meant "right to left" in every case, because you already know the weight of the righmost digit (it is always 1) but not the weight of the leftmost digit until we count digits ... So i suggested to go from right to left.

• ###### 13. Re: Binary And Decimal

When We Say the 1st is the blue circle ?

• ###### 14. Re: Binary And Decimal

The formal/technical term for the red part, in binary, is most significant bit and in decimal it is most significant digit.  The blue position, in binary, is the least significant bit and in decimal it is the least significant digit.

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