# 0.2 Essential mathematics

 Page 1 / 2

## Exponential arithmetic

Exponential notation is used to express very large and very small numbers as a product of two numbers. The first number of the product, the digit term , is usually a number not less than 1 and not greater than 10. The second number of the product, the exponential term , is written as 10 with an exponent. Some examples of exponential notation are:

$\begin{array}{ccc}\hfill 1000& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \hfill 100& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}\hfill \\ \hfill 10& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{1}\hfill \\ \hfill 1& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{0}\hfill \\ \hfill 0.1& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}\hfill \\ \hfill 0.001& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \\ \hfill 2386& =& 2.386\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1000=2.386\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \hfill 0.123& =& 1.23\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.1=1.23\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}\hfill \end{array}$

The power (exponent) of 10 is equal to the number of places the decimal is shifted to give the digit number. The exponential method is particularly useful notation for every large and very small numbers. For example, 1,230,000,000 = 1.23 $×$ 10 9 , and 0.00000000036 = 3.6 $×$ 10 −10 .

Convert all numbers to the same power of 10, add the digit terms of the numbers, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

Add 5.00 $×$ 10 −5 and 3.00 $×$ 10 −3 .

## Solution

$\begin{array}{ccc}\hfill 3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}& =& 300\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\hfill \\ \hfill \left(5.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)+\left(300\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)& =& 305\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}=3.05\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}$

## Subtraction of exponentials

Convert all numbers to the same power of 10, take the difference of the digit terms, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

## Subtracting exponentials

Subtract 4.0 $×$ 10 −7 from 5.0 $×$ 10 −6 .

## Solution

$\begin{array}{}\\ 4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}=0.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\\ \left(5.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)-\left(0.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)=4.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\end{array}$

## Multiplication of exponentials

Multiply the digit terms in the usual way and add the exponents of the exponential terms.

## Multiplying exponentials

Multiply 4.2 $×$ 10 −8 by 2.0 $×$ 10 3 .

## Solution

$\left(4.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\right)=\left(4.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2.0\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\left(-8\right)+\left(+3\right)}=8.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$

## Division of exponentials

Divide the digit term of the numerator by the digit term of the denominator and subtract the exponents of the exponential terms.

## Dividing exponentials

Divide 3.6 $×$ 10 5 by 6.0 $×$ 10 −4 .

## Solution

$\frac{3.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}}{6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}}\phantom{\rule{0.2em}{0ex}}=\left(\frac{3.6}{6.0}\right)\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\left(-5\right)-\left(-4\right)}=0.60\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}=6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

## Squaring of exponentials

Square the digit term in the usual way and multiply the exponent of the exponential term by 2.

## Squaring exponentials

Square the number 4.0 $×$ 10 −6 .

## Solution

${\left(4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)}^{2}=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(-6\right)}=16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}$

## Cubing of exponentials

Cube the digit term in the usual way and multiply the exponent of the exponential term by 3.

## Cubing exponentials

Cube the number 2 $×$ 10 4 .

## Solution

${\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\right)}^{3}=2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}4}=8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{12}$

## Taking square roots of exponentials

If necessary, decrease or increase the exponential term so that the power of 10 is evenly divisible by 2. Extract the square root of the digit term and divide the exponential term by 2.

## Finding the square root of exponentials

Find the square root of 1.6 $×$ 10 −7 .

## Solution

$\begin{array}{ccc}\hfill 1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}& =& 16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\hfill \\ \hfill \sqrt{16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}}=\sqrt{16}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\sqrt{{10}^{-8}}& =\hfill & \sqrt{16}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-\phantom{\rule{0.2em}{0ex}}\frac{8}{2}}\phantom{\rule{0.2em}{0ex}}=4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\hfill \end{array}$

## Significant figures

A beekeeper reports that he has 525,341 bees. The last three figures of the number are obviously inaccurate, for during the time the keeper was counting the bees, some of them died and others hatched; this makes it quite difficult to determine the exact number of bees. It would have been more accurate if the beekeeper had reported the number 525,000. In other words, the last three figures are not significant, except to set the position of the decimal point. Their exact values have no meaning useful in this situation. In reporting any information as numbers, use only as many significant figures as the accuracy of the measurement warrants.

What is rightful definition of element
is alkanes a saturated hydrocarbon?
yup
it's saturated cos it has single bonds
yh....because they don't undergo additional reactions which hydrogen and other atoms can add across the carbon-carbon or triple bond
patience
and me...I'm I wrong?
patience
how does metal looses electron
By oxidation and reduction
hamidat
by oxidation loss
Official
An acid is a proton donor.
what is an acid
an acid is a substance when dissolved in water produces hydrogen ion or hydroxonium ion
hamidat
good
Mudassir
thanks
hamidat
what is ionic bonding
It involves the transferring of electron from a metal to a non mental
hamidat
that's right
Edward
bonding between a metal and a non metal
miriam
calculate the hydrogen ion concentration of the solution when pH=5
no ideas
hamidat
What is thermodynamics
what is the meaning this word twentieth
dhu
Is the branch of physics that deal with heat and temperature and their relation to work, energy and properties of matter
Edward
There are no topics on hydrocarbons
I don't understand
hamidat
its not making sense to me I still don't understand
How and why
Betrice
yes
megan
we need diagram for easy going and understand
serah
How can we easily differentiate between the 5 gas laws
how many carbon is present in alkene
it's the carbon to carbon being double bonded to each other that makes it an alkene, not the amount of carbon itself. ex: C=C, C=C=C. both are alkenes.
Phill
I need more light on alkene
chidera
other usefullness of hydrogen apart from this, it is colourless, odourless and tasteless
it is neutral to litmus paper, it is insoluble in water
hamidat
the enthalpy of a system
changing in heat of a system which can be lost or gained
Yussuf
what is the unit of pressure
what is pressure measured in?
Tim
Pascal
Stupid
millimeter mercury ,mmHg or ATM
hamidat