I fail to understand the significance or perhaps visualize the use of the multiplicative inverse =
$1\over n$
What is the use? And how it can help in problems?
Update: Mainly, my question was why would you multiply by a multiplicative inverse when you could divide by the coefficient? To which the answer is the Babylonians didn't have a long division algorithm.
On
The qualifier multiplicative is used to avoid confusion with the additive inverse (which yields the opposite rather than the reciprocal).
The inverse is of a number is such that when it multiplies that number, the product is the multiplicative neutral.
$$n\times i=1.$$
For example, in modulo $5$ arithmetic,
$$1\times1=1,\\2\times3=1,\\3\times2=1,\\4\times4=1.$$
$0$ has no inverse.
You can use it to solve equations like $a\times x=b$.
On
The following is a list of properties of the multiplicative inverse.
1) Every non-zero real or complex number has a unique multiplicative inverse.
2) If you multiply a number by its multiplicative inverse you get number $1$
3) Multiplicative inverse of multiplicative inverse of a number is the number itself.
4) Multiplicative inverse of positive numbers are positive and multiplicative inverse of negative numbers are negative.
5) As numbers gets smaller and smaller, their multiplicative inverse gets larger and larger.
6) Zero does not have a multiplicative inverse.
7) Dividing two numbers is the same as multiplying the first number by the multiplicative inverse of the second number.
8) There are only two real numbers whose multiplicative inverse equals themselves. They are $1$ and $-1$
9) Multiplicative inverse of $10,100,1000,10000,....$ are $0.1, 0.01, 0.001,0.0001,.....$ and vice versa.
10) $1-1/2+1/3-1/4+.... = ln(2)$
11) $1-1/3+1/5 -1/7 +1/9.....= \frac {\pi }{4}$
12) $1+1/2+1/3+1/4+.... = \infty $
And many many more.
On
In the very abstract sense is something called group with respect to multiplication has following property:
1) For all element a in Group G there exist unique element b in Group G such that a.b=1 where . is multiplication operation and 1 is identity in that group G.
Common example is group of rationals: For every rational number a there exist unique another rational such that a.b=1
This is not case for integers since 2*1/2=1 but 1/2 is not an integer hence integer is not group.
Same we can do with matrices whose determinant is non zero. We know every non zero determinant matrix has unique inverse so it is multiplicative inverse where identity of this type of matrices group is simply identity matrix.
On
Inverses undo operations. Additive inverses undo additions, e.g., to undo $x + 7$, you use $-7$, the inverse of 7 thus: $x - 7$. In solving an equation, you might arrive at a point where you have something like $5x + 7 = 52$. Using the additive inverse gets you one step closer to the answer: $5x = 45$.
Likewise with the multiplicative inverse, it undoes a multiplication. The multiplicative inverse of 5 is $\frac{1}{5}$. Multiplying both sides of the example equation by $\frac{1}{5}$ gets us the answer: $x = 9$.
Maybe that's not the best example. The multiplicative inverse, or reciprocal, kinda gets taken for granted there. Reciprocals are also useful in some divisions when you want or have to do them on paper or in your head. For example, what's one seventh of three quarters? $$\frac{\frac{3}{4}}{7} = \frac{3}{4} \times \frac{1}{7} = \frac{3 \times 1}{4 \times 7} = \frac{3}{28}.$$
A slightly more interesting example: what's two sevenths of three quarters? $$\frac{\frac{3}{4}}{\frac{2}{7}} = \frac{3}{4} \times \frac{7}{2} = \frac{3 \times 7}{4 \times 2} = \frac{21}{8}.$$
A multiplicative inverse is a reciprocal.
A reciprocal is one of a pair of numbers that when multiplied with another number equals the number $1$. For example, if we have the number $7$, the multiplicative inverse, or reciprocal, would be $\cfrac 17$ because when you multiply $7$ and $\cfrac 17$ together, you get $1$!
When you divide a number by another number, you actually multiply by the inverse of the divisor .
$\cfrac 75 = 7 × \cfrac 57$
The multiplicative inverse is very significant in algebra, where you can only divide by multiplying the reciprocal of the divisor.