3. Operations with Integers, Fractions & Decimals
Some simple rules for signs:
The negative of a negative is positive. e.g. $-(-3)=3; 3-(-2)=3+2=5$
The product of two negatives is a positive. e.g. $(-3)*(-2)=6$
The product of a positive and a negative is a negative. e.g. $(-3)*2=(-6)$
The same rules above hold for division as well.
And the same rules hold for all of integers, decimals, fractions etc.,
Some simple rules for even and odd numbers:
odd times even gives an even
odd times odd gives an odd
even times even gives an even
multiplication is commutative with real numbers, so an odd times an even gives the same results if the ordering of the numbers is reversed.
If x and y are integers, a number expressed as x/y is called a fraction, with numerator x and denominator y.
If x and y are each expressed as the product of their prime factors, and common prime factors cancelled from the numerator and denominator, then the fraction is in its reduced form.
A fraction with denominators like $10, 100 (=10*10), 1000 (=10*10*10)$, ... can be converted into a decimal form.
This is done by placing a decimal point in the numerator counting right to left, after the number of zeros in the denominator.
e.g. $3/10=0.3. 233/100=2.33. 3/1000=0.003$.
The first digit to the right of the decimal point is the tenths place, then the 100ths place and so on.
This is akin to how the places to the left of the decimal point are the units, tens, hundreds etc from right to left.
The value of a fraction x/y does not change if both numerator and denominator are multiplied by the same quantity.
We can use this idea to convert a fraction to a decimal, and vice versa.
e.g. $8/125=(8*8)/(125*8)=64/1000-0.064$ (Recall that you need the denominator to be 10, 100, 1000, ... for the conversion to work)
e.g. $0.064=0.64/10=6.4/100=64/1000$ (Recall that each 10 in the denominator pushes the decimal point one place to the left in the numerator.
Some decimals can have recurring patterns of digits. These are digits after the decimal points that continue on... and on.
e.g. $3.34343434....$ to $∞$ how to express this decimal as a fraction?
Let's say $p=3.3434343434...$ to $∞$ (call this A). Then $100*p=334.34343434...$ to $∞$ B
So subtracting equation B from equation A, we get, $99p=331$, or $p=331/99$
To add (or subtract) two fractions, we need to normalize their denominators first.
We cannot add the numerators of two fractions until their denominators are equal. Normalizing the denominators achieves this.
e.g. to add $a/b$ and $c/d$, we multiply both the numerator and denominator of the first fraction by $d$.
Then we multiply the numerator and denominator of the second fraction by $b$.
So we get two fractions $(a*d)/(b*d)$ and $(c*b)/(b*d)$.
The values of the fractions haven't changed since $(d/d)=1$ and $(b/b)=1$, we effectively just multiplied each fraction by 1.
So, $a/b + c/d = (ad)/(bd) + (bc)/(bd) = (ad+bc)/(bd)$.
Multiplication of two fractions does not require normalizing the denominators.
We just multiply the two numerators and the two denominators to get the resultant fraction
Division of two fractions is the same as multiplying the first fraction with the second fraction's reciprocal.
To add (or subtract) two decimals, we need to align digits so we add or subtract corresponding digits in the right places.
Operations proceed normally - like regular addition and subtraction otherwise
e.g. $123.321 - 23.32 = 100.001$
e.g. $23.32 + 110.011 = 133.331$
To multiply two decimals, simply multiply the two numbers ignoring the decimal point, then put it back in summing the number of digits to the right of the decimal point in the two numbers.
e.g. $23.21*12.2=2321*122/(1000)=283.162$ (2 decimals+1 decimal=3 digits after decimal point in the product).
The above logic works in reverse for divisions.