# 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