Real Numbers – HSC Math

Posted In: Mathematics

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. Real number set can be represented by the symbol R.

  1. If $a>b$ and $(x-a)(x-b)<0$ then $x<a$ and $x>b$.
    So $b < x < a$.
  2. If $a>b$ and $(x-a)(x-b)>0$ then $x>a$ or $x<b$.
    So $x < b$ or $x > a$.
  3. $|x|=\{\table x\ \when\ x≥0;-x\ \when x<0$
  4. $|x|<α,$ if and only if, $-α<x< α$
  5. $|x|≥x$
  6. $|x|^2=x^2$
  7. $|ab|=|a||b|$
  8. $|a+b|≤|a|+|b|$
  9. $|a|-|b|≤|a-b|$
  10. $-|a|≤a≤|a|$
  11. Axoims of Real Numbers

  12. Closure:
    If a,b∈R then
    i. $a+b∈R$
    ii. $ab∈R$
    iii. $a-b∈R$
    iv. $a/b∈R \ (b≠0)$
  13. Commutativity:
    If a,b∈R then
    i. $a+b=b+a$
    ii. $ab=ba$
  14. Associativity:
    If a,b,c∈R then
    i. $(a+b)+c=a+(b+c)$
    ii. $(ab)c=a(bc)$
  15. Distributivity:
    If a,b,c∈R then
    i. $a(b+c)=ab+ac$
    ii. $(a+b)c=ac+bc$
  16. Existence of Identity:
    If a∈R then
    i. $a+0=0+a$
    ii. $a.1=1.a$
  17. Existence of Inverse:
    If a∈R then
    i. $a+(-a)=(-a)+a$
    ii. $a.a^1=a^1.a$
  18. Uniqueness:
    If a,b,c,d∈R and a=b and c=d then
    i. $a+c=b+d$
    ii. $ac=bd$

Post Tags: HSC | Number Sets


No Comments »


Leave a Reply

Your email address will not be published. Required fields are marked *