**Q. Illustrate Field Properties of Numbers?**

**Ans.**

What the **associative law of addition** states is this: for any numbers a, b, and c,

(*a* + *b*) + *c* = *a* + (*b* + *c*).

There's a similar **associative law for multiplication**:

(*ab*)*c* = *a*(*bc*).

The associative laws are about the way in which the numbers can be associated (grouped together). The commutative laws are about the way in which numbers can be moved around in an expression. The **commutative law for addition **is:

*a* + *b* = *b* + *a*;

and **for multiplication**:

*ab* = *ba*.

The **distributive law **shows how addition and multiplication are related:

*a*(*b* + *c*) = *ab* + *ac*.

All these laws work for any real or complex number, and they even work for entire expressions. For example, by the commutative law for multiplication,

(*5x* + *1*)*y* = *y*(*5x* + *1*).

**For advanced students**: Just because these laws work for numbers, don't assume that they work for other things, like matrices! If A and B are matrices, it's true that

*A* + *B* = *B* + *A*,

but it's **not** necessarily true that

*AB* = *BA*.