Scientific maths help

[ScienceNoobie]

Hello, as my username suggests... I am new to science and I am studying online so it is a bit harder to get help with my questions. 

I have answered the following sum but I have 3 possible answers (probably wrong) so i just want to confirm how to actually work this out properly to see if any of my answers are correct.

Also how should I properly display the answer if asked to "give your answer in scientific notation and without any brackets" does this mean show my answer as a sum such as this:

A x 10 to the power of B?

(not sure how to write in small nexto the 10 to shower the power properly but I'm sure you understand what I mean above) 

[Strange]

3 hours ago, ScienceNoobie said:

I have answered the following sum but I have 3 possible answers (probably wrong) so i just want to confirm how to actually work this out properly to see if any of my answers are correct.

You could post your three answers and we might be able to tell you which is right (if one is  ) and why you have got the others wrong.

What makes scientific notation so nice is that you only have to divide the mantissa (the first bit of the number) and you can subtract the exponents (the "10 to the power" bit). So in the case, you divide 4.80 by 2.50 and subtract -26 from -14.

3 hours ago, ScienceNoobie said:

A x 10 to the power of B?

Exactly.

You can get the superscript from one of the buttons above the editor box (maybe not if you are on a phone). Another standard way of writing 10¹²,for example, is 10^12

[studiot]

4 hours ago, ScienceNoobie said:

Hello, as my username suggests... I am new to science and I am studying online so it is a bit harder to get help with my questions. 

 

Plenty of help available here for those that are interested.

 

A bit more about this subject that might be useful.

 

We say that a number (say a)  multiplied by itself n times is "a raised to the power n"

we write it thus aⁿ.

This site is unusual in that the tools along the top of the entry box allow superscript X²  (and also subscript) X₂. Both of these are much used in technical writing.

n is also called the exponent, as Strange says.

But beware he has supplied the computing definition of "mantissa", which is different from the mathematical one.
So I would avoid that term until you know more.

Anyway powers of 10 are special because they move the decimal point of a number thus:

2348.56 = 234.856 x10⁰ = 23.4856 x 10¹ = 2.34856 x 10² = 0.234856 x10³ = 2348.56 x10^-1 = 23485.6 x 10^-2 = 234856 x 10^-3.

So with this notation we can get rid of the decimal point altogether.
But this is at the expense of having to write a superscript.
Note also that I have used x 10⁰. Anything to the power 0 is equal to1

So some calculators and computers may use an alternative method which can be written all along one line.
This is called the E or e notation (after exponent, remember them?)
So my example above may appear on a calculator like this

234.856 = 234.856 = 23.4856E1 = 2.34856E2 = 0.234856E3 = 2348.56E-1 = 23485.6E-2 = 234856E-3.

Here is a little webapp to play with.

https://www.calculatorsoup.com/calculators/math/scientific-notation-converter.php

 

The point of all this is to use a format that helps avoid arithmetical errors.

Suppose you have a problem to work out that leads to the expression

[math]\frac{{0.0035\;x\;56892}}{{120\;x\;100}}[/math]

We can rewrite this expression using powers of 10

[math]\frac{{3.5\;x\;{{10}^{ - 3}}\;x\;5.6892\;x\;{{10}^4}}}{{1.2\;x\;{{10}^2}\;x\;1\;x\;{{10}^2}}}[/math]

Some of the powers of 10 can be cancelled out.
This is a very good reason for writing the fraction as I have done, not using the division sign.

[math]\frac{{3.5\;x\;5.6892}}{{1.2}}\;x\;{10^{ - 3}}[/math]

Now all the numbers to multiply or divide are in the range 0 - 10.
Wew should first divide the 3.5 by the 1.2 to keep it that way.
Then multiply the result by the 5.6892.
This maintains the best possible accuracy.
Note also that I have collected together all the remaining power of 10 (-3 in this case) and moved them out of the fraction.

[math]16.5935\;x\;{10^{ - 3}}[/math]

Again going to a number between 0 and 10 and a power of 10 we have

[math]1.65935\;x\;{10^{ - 2}}[/math]

 

Hope this helps

 

[Strange]

2 hours ago, studiot said:

But beware he has supplied the computing definition of "mantissa", which is different from the mathematical one.

What is the mathematical name for that bit of the number? (I wasn't sure that mantissa was right, but it was the only thing that came to mind!)

[studiot]

1 hour ago, Strange said:

What is the mathematical name for that bit of the number? (I wasn't sure that mantissa was right, but it was the only thing that came to mind!)

mantissa

/manˈtɪsə/

Learn to pronounce

noun

noun: mantissa; plural noun: mantissas

1. 1.

   Mathematics

   the part of a logarithm after the decimal point.

    
2. 2.

   Computing

   the part of a floating-point number which represents the significant digits of that number.

    

Origin

mid 17th century: from Latin, literally ‘makeweight’, perhaps from Etruscan.

 

Translate mantissa to

 

Use over time for: mantissa

 

Translations, word origin and more definitions

From Oxford

 

 

Sorry I see  an error in my list

2348.56 = 234.856 x10⁰ = 23.4856 x 10¹ = 2.34856 x 10² = 0.234856 x10³ = 2348.56 x10^-1 = 23485.6 x 10^-2 = 234856 x 10^-3.

This should be

234.856 = 234.856 x10⁰ = 23.4856 x 10¹ = 2.34856 x 10² = 0.234856 x10³ = 2348.56 x10^-1 = 23485.6 x 10^-2 = 234856 x 10^-3.

 

 

[Strange]

55 minutes ago, studiot said:

mantissa

/manˈtɪsə/

Learn to pronounce

noun

noun: mantissa; plural noun: mantissas

1. 1.

   Mathematics

   the part of a logarithm after the decimal point.

    
2. 2.

   Computing

   the part of a floating-point number which represents the significant digits of that number.

    

Origin

mid 17th century: from Latin, literally ‘makeweight’, perhaps from Etruscan.

 

Translate mantissa to

 

Use over time for: mantissa

 

Translations, word origin and more definitions

From Oxford

Errrr.... thanks. I think. (It doesn't really answer my question, though.)

[studiot]

24 minutes ago, Strange said:

Errrr.... thanks. I think. (It doesn't really answer my question, though.)

Guessing you mean the 3 in   3 x 10⁵  I have not heard one either.

 

Google gives this, which is slightly more restrictive  than my (engineering) version.

 

Quote

How It Works

Scientific notation has three parts to it: the coefficient, the base, and the exponent.

The coefficient must be greater than 1 and less than 10 and contain all the significant (non-zero) digits in the number.

• 12.5 × 10⁶ is not in proper scientific notation, since the coefficient is greater than 10.
• Neither is 0.125 × 10⁷, since the coefficient is less than 1.

The base is always 10.

The exponent is the number of places the decimal was moved to obtain the coefficient.

 

https://www.shmoop.com/study-guides/pre-algebra/basic-operations/scientific-notation

[Strange]

This page: https://www.mathsisfun.com/numbers/scientific-notation.html (which might be useful to the OP - and apologies if it is too basic) just calls it "the digits".

Quote

After putting the number in Scientific Notation, just check that:

• The "digits" part is between 1 and 10 (it can be 1, but never 10)
• The "power" part shows exactly how many places to move the decimal point

It also goes on to describe engineering notation which, as an engineer, I have always found very confusing.

Wikipedia gives three names for it: coefficient, significand and mantissa.

Quote

In scientific notation, all numbers are written in the form

m × 10ⁿ

(m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number. The integer n is called the order of magnitude and the real number m is called the significand or mantissa.^[1]

https://en.wikipedia.org/wiki/Scientific_notation

Although the discussion of the naming is slightly off topic, hopefully the links are useful to the OP... (but we had better drop it now!)

[Carrock]

Quote

The exponent is the number of places the decimal was moved to obtain the coefficient.

Use Maxwell's middle hand rule to determine the sign of the exponent. Exponent size is just a matter of adding or subtracting something ...

etc.

Ultraoversimplified.....

 

[ScienceNoobie]

Thank you guys all for the help, I got the answer correct after reading all your replies... I didn't realise it could be so easy as to divide the full numbers then just subtract the powers and write that in a scientific fomat. 

Thank you again all for the help.