Purpose of using 10^7 or any # raised

[Niaboc67]

This might be a silly question and an obvious one, it's something I just wanted confirmation/clarification on. I've been recently reading a lot of mathematical papers/journals from around the world and have noticed something in particular. A lot of formulas or things they are postulating use raised powers. For example 10^7 which of course is the same as 10000000. Now is this just for connivence sake? couldn't they just use 10000000 instead? Or is there some property held within writing things in raised form?

Thank you

 

[NascentOxygen]

Convenience, yes. But really large and really small numbers become just too unwieldy. It's easy to miscount in a string of zeros if you are doing calculations, maybe needing to cancel some powers of 10 in the numerator with some in the denominator. Writing it as 10⁷ means you don't have to count up those 7 zeros---the counting is already done.

6.24 x 10¹⁸ electronic charges = 1 coulomb, and that's a lot of 0's to keep writing out!

Avogadros number is 6.02 x 10²³

 

[Niaboc67]

@NascentOxygen wow...that just really opened my eyes to a lot. So just take the number of arbitrary things (that you already know the value of) and then 10^18 or whichever exponent would be the most efficient way of that type of calculation. What about other bases? that do not look so clean? like 7^7 or 6^6. I can't say that I see those. Are these unlikely to be seen in formulas ever? why is it generally base 10 instead of any other numbers? If a calculation were to be more intricate would they just use the number instead like like for example:14485792854 because thee isn't a way to rewrite that with base 10 or a neater way of forming that?

Thanks!

 

[SteamKing]

@NascentOxygen wow...that just really opened my eyes to a lot. So just take the number of arbitrary things (that you already know the value of) and then 10^18 or whichever exponent would be the most efficient way of that type of calculation. What about other bases? that do not look so clean? like 7^7 or 6^6. I can't say that I see those. Are these unlikely to be seen in formulas ever? why is it generally base 10 instead of any other numbers? If a calculation were to be more intricate would they just use the number instead like like for example:14485792854 because thee isn't a way to rewrite that with base 10 or a neater way of forming that?

Thanks!

You can use exponential notation in other bases. It's mostly used in base 10 because that is the base in which most quantities are expressed and the base which we use for mathematics.

In computer science, base 2, or binary, is used commonly because the elements there are either on (1) or off (0). To help manage an unwieldy amount of 1s and 0s, other forms of binary are commonly used, such as hexadecimal (base 16) or octal (base 8), because 16 and 8 share a factor of 2.

For example, if you have the binary number 1101 1110 0101, it can be expressed as 110 111 100 101 or 6745 base 8, or DE5 in base 16, or 3557 in base 10.

 

[NascentOxygen]

why is it generally base 10 instead of any other numbers? If a calculation were to be more intricate would they just use the number instead like like for example:14485792854 because thee isn't a way to rewrite that with base 10 or a neater way of forming that?

It's powers of ten because each 0 you append represents multiplying by a factor of 10.

14485792854 can be written as 1.4485792854 x 10¹⁰

It is standard notation to write quantities as having a single digit left of the decimal point, then multiplied by the appropriate power of 10. Numbers smaller than 1 will be shown having a negative power of 10, e.g.,
0.00000456 = 4.56 x 10^-6

 

[mafagafo]

6.24 x 10¹⁸ electronic charges = 1 coulomb, and that's a lot of 0's to keep writing out!

Avogadros number is 6.02 x 10²³

Or even worse: Planck's constant!

Also: 10^7 saves space! If you are going to repeat a number 10 times, it matters if it has 20 or 5 digits!

 

[Mark44]

You can use exponential notation in other bases. It's mostly used in base 10 because that is the base in which most quantities are expressed and the base which we use for mathematics.

In computer science, base 2, or binary, is used commonly because the elements there are either on (1) or off (0). To help manage an unwieldy amount of 1s and 0s, other forms of binary are commonly used, such as hexadecimal (base 16) or octal (base 8), because 16 and 8 share a factor of 2.

For example, if you have the binary number 1101 1110 0101, it can be expressed as 110 111 100 101 or 6745 base 8, or DE5 in base 16, or 3557 in base 10.

To add to what SteamKing said, floating point numbers are stored in a computer in a sort of scientific notation, except that the base is 2 instead of 10. In this representation a fractional part and an exponent on 2 are stored. For example, the decimal fraction .1875 (= 3/16) is stored as something like 1.1 X 2^-3, which is the same as 0.0011₂. This binary fraction indicates 0 X 1/2 + 0 X 1/4 + 1 X 1/8 + 1 X 1/16.

 

[Niaboc67]

Ok, now I have even more question than I started with here lol. Hopefully each one of you can clear them up. I'll start in order of who answered when:

@SteamKing
"It's mostly used in base 10 because that is the base in which most quantities are expressed and the base which we use for mathematics"
Is this because it's what mathematics is use to using tradionally? or if we were to use other number it would never look as neat as 10. Which this brings me to another basic question about mathematics why does 10 have the property to make things neat and clean. Is it due to it's association with 1,10,100,1000 with the beginning of all numbers. I am aware that this is like probably something a first grader would ask but this was never properly explained to me and I've gone through a terrible educational system.

What exactly does hexadecimal (base 16) mean? Is that like saying you are representing a set of 16 digits of 1's and 0's?
And with hexadecimal (base 16) and octal (base 8) could we not ramp this up further with say 32,64,128 or all numbers which are divisible by two? Or might there be a sort of computational limit here?

"For example, if you have the binary number 1101 1110 0101, it can be expressed as 110 111 100 101 or 6745 base 8, or DE5 in base 16, or 3557 in base 10"
Sorry but I am not understanding that. What is happening to the 1101 1110 0101 to be able to express it as 110 111 100 101? And 8^6745 I am a bit lost with that sentence.

@NascentOxygen
"14485792854 can be written as 1.4485792854 x 1010"
In a case such as this would it not be easier to write it as the former? but is it more mathematical tradition?
0.00000456 = 4.56 x 10-6
Out of curiosity when would a number such as this be needed? Just an example.

@Mark44
"To add to what SteamKing said, floating point numbers are stored in a computer in a sort of scientific notation, except that the base is 2 instead of 10. In this representation a fractional part and an exponent on 2 are stored. For example, the decimal fraction .1875 (= 3/16) is stored as something like 1.1 X 2-3, which is the same as 0.00112. This binary fraction indicates 0 X 1/2 + 0 X 1/4 + 1 X 1/8 + 1 X 1/16."

The reason number are stored as either one or zero is because transistors/logic gates can only be on and off, is that correct? Therefore we had to devise an ever complex system of working with 1's and 0's? I am just a bit confused as to how a computer processes decimal fractions as a form of 1's and 0's. When you write 0 X 1/2 + 0 X 1/4 + 1 X 1/8 + 1 X 1/16 is still considered binary to the computer? THANKS

 

[NascentOxygen]

In our base 10 number system we use digits 0-9, and each "column" is worth 10 times the column to its right, e.g.,
1000 100 10 1 . 1/10 1/100

So a number 2506 _{(base 10)} is worth 2x1000+5*100+0*10+6*1

In a base 3 number system you use digits 0-2, only, and each column is worth 3 times the column to its right, e.g.,

81 27 9 3 1 . 1/3 1/9

So a number 20112.2 _{(base 3)} is worth 2*81+0*27+1*9+1*3+2*1+2*1/3
and you can use your calculator to work out what this will be in decimal.

@NascentOxygen
"14485792854 can be written as 1.4485792854 x 1010"
In a case such as this would it not be easier to write it as the former? but is it more mathematical tradition?
0.00000456 = 4.56 x 10-6
Out of curiosity when would a number such as this be needed? Just an example.

Yes, exact integers are often written as a big number with no decimal point where it's important that that they stay exact in computation. If that number represented the population of a country, it would probably be rounded off as, say, 14.49 trillion, or 14.49 x 10⁹ or 1.449 x 10¹⁰

That small number 4.56 x 10^-6 might be the thickness of a film of plastic, in metres. Or the thickness of a layer of gold plated onto a metal ring or ornament.

Another example:
-1.602 x 10^-19 coulombs is the electric charge on a single electron.

http://thumbnails112.imagebam.com/37333/0363e9373324851.jpg

 

[SteamKing]

Ok, now I have even more question than I started with here lol. Hopefully each one of you can clear them up. I'll start in order of who answered when:

@SteamKing
"It's mostly used in base 10 because that is the base in which most quantities are expressed and the base which we use for mathematics"
Is this because it's what mathematics is use to using tradionally? or if we were to use other number it would never look as neat as 10. Which this brings me to another basic question about mathematics why does 10 have the property to make things neat and clean. Is it due to it's association with 1,10,100,1000 with the beginning of all numbers. I am aware that this is like probably something a first grader would ask but this was never properly explained to me and I've gone through a terrible educational system.

The number system currently used, which is called the decimal, or base 10, system is able to represent a given number with a few symbols because it is a positional system, where the order in which the numerals are written also imply multiplication by powers of the base.

For example, 3557 is a shorthand way of representing 3*1000 + 5*100 + 5*10 + 7*1. The powers of 10, namely 1, 10, 100, and 1000 are suggested by the location of each digit in 3557.

Other number systems, like Roman numerals, don't use a consistent positional representation with a common base. For example, XL in Roman numerals means you subtract 10 (X) from 50 (L) to make 40 (XL). If, on the other hand, you want to write 43 in Roman numerals, it would be XLIII, where the three Is are added to the XL by being written to the right.

Positional representation of numbers not only makes for a compact expression, but doing arithmetic is also greatly simplified.

What exactly does hexadecimal (base 16) mean? Is that like saying you are representing a set of 16 digits of 1's and 0's?

No. A numeric base is how many distinct symbols are used to represent any number, in a positional format.

For example, in the decimal, or base 10, system, there are ten distinct signs for individual numerals: 0 1 2 3 4 5 6 7 8 9
Any number in this system can be represented by a combination of one or more of these numerals written in a certain order.
For example, the number 3557 indicates 3*10³ + 5*10² + 5*10¹ + 7*10⁰
Note how the exponent of the base (10) increases the further left one goes from the (implied) decimal point.

In the hexadecimal, or base 16, system, there are 16 distinct numeral signs: 0 1 2 3 4 5 6 7 8 9 A B C D E F
The letters A-F are used here because there are no other numeral signs with which to represent these values as a single symbol. Arbitrary symbols could have been devised, but the first six letters of the alphabet were chosen instead.

Thus:
A base 16 = 10 base 10
B base 16 = 11 base 10
C = 12
D = 13
E = 14
F = 15

Going back to our example, 6745 in base 10 is also equal to DE5 in base 16, or
D * 16² + E * 16¹ + 5 * 16⁰ = 13*256 + 14*16 + 5*1 = 3328 + 224 + 5 = 3557 base 10

And with hexadecimal (base 16) and octal (base 8) could we not ramp this up further with say 32,64,128 or all numbers which are divisible by two? Or might there be a sort of computational limit here?

It's not clear what you are talking about here.

"For example, if you have the binary number 1101 1110 0101, it can be expressed as 110 111 100 101 or 6745 base 8, or DE5 in base 16, or 3557 in base 10"
Sorry but I am not understanding that. What is happening to the 1101 1110 0101 to be able to express it as 110 111 100 101? And 8^6745 I am a bit lost with that sentence.

What I was trying to show is that by grouping the digits in a binary number, it is very easy to convert the binary number into a base 8 or base 16 representation very easily.

Binary format: 110111100101

By grouping the binary digits in threes, it makes it very easy to find the base 8, or octal, form:

110111100101 → 110 111 100 101 → 6 7 4 5 → 6745 octal → 3557 decimal

By grouping the binary digits in fours, it makes it very easy to find the base 16, or hexadecimal, form:

110111100101 → 1101 1110 0101 → D E 5 → DE5 hexadecimal → 3557 decimal

 

[Mark44]

And with hexadecimal (base 16) and octal (base 8) could we not ramp this up further with say 32,64,128 or all numbers which are divisible by two?

Yeah, you could. In fact, base-64 is used quite often in Web transactions. I don't know of base-32 being used or base-128, but it's possible that someone will find a need for higher bases.

Or might there be a sort of computational limit here?

 

[SteamKing]

Yeah, you could. In fact, base-64 is used quite often in Web transactions. I don't know of base-32 being used or base-128, but it's possible that someone will find a need for higher bases.

I think the base 64 scheme referred to here is primarily used as a coding method for text, rather than for use in numeric computations:

http://en.wikipedia.org/wiki/Base64

The octal and hexadecimal representation of binary numbers was devised as a means of expressing large binary numbers in a convenient format for human to process.

Instead of a human having to read and understand a 16-bit binary number, especially in printed documentation, it was converted to a string of 4 hexadecimal digits. The computer still processes and stores the 16-bit binary number internally.

 

[SteamKing]

And with hexadecimal (base 16) and octal (base 8) could we not ramp this up further with say 32,64,128 or all numbers which are divisible by two? Or might there be a sort of computational limit here?

The problem with using a number system with a base of 32, 64, or 128 is that it becomes rather unwieldy in use.

For example, with a base 32 system, you must have 32 distinct numeral signs in order to represent numbers. Like the hexadecimal system, we could use the ten numeral signs for representing numbers in base 10, but that leaves us 22 signs short. If we use letters of the alphabet, then that means we have to learn what each letter represents numerically. Thus:

0 1 2 3 4 5 6 7 8 9 Plus
A - 10
B - 11
C - 12
D - 13
E - 14
F - 15
G - 16
H - 17
J - 18
K - 19
L - 20
M - 21
N - 22
P - 23
R - 24
S - 25
T - 26
U - 27
V - 28
X - 29
Y - 30
Z - 31
[Note: I and O are omitted here because they could be confused with the numerals 0 (zero) and 1 (one).]

You can see that it requires a lot of memorization to learn the values of each symbol, and no gains would be realized from a computational standpoint by using such a large numeric base.

 

[Jonathan Scott]

I think the base 64 scheme referred to here is primarily used as a coding method for text, rather than for use in numeric computations

Base 64 is used for representing general binary data (6 bits at a time) as common printable text characters, in such a way that if the text is converted to some other representation (for example ASCII to EBCDIC translation) the binary data will still be preserved. The same idea applies for example to binary, decimal and hexadecimal numbers represented in text form, but base 64 is more efficient in representing more bits per character.

 

[DrClaude]

One thing that wasn't mentionned about scientific notation is its use to specify the number of significant digits. If you measure something as being 9.9 km, but need to report it in meters, writing 9900 m leaves an ambiguity as to how many significant digits there are. Was the mesurement precise to the nearest meter? Writing 9.9 × 10³ m makes it clear that there are only two significant digits.