Math 10

Real number

Rational numbers

Irrational numbers

π , √2

Integers

Whole

Natural

12

Rational Like: 34, 52,12,23

Integers {…, -3, -2, -1, 0, 1, 2, 3…….}

Whole {0, 1, 2, 3…}

Natural {1, 2, 3…}

Properties of real numbers

1- Reflexive property a = a

2- Symmetric property a = b then b = a

3- Transitive property a = b and b = c then a = c

4- Principle of substitution if a = b then we can substitute b for a in any expirations

Commutative properties

a + b = b + a , a . b = b . a

Associative properties

a + ( b + c ) = ( a + b ) + c = a + b + c

a . ( b . c ) = ( a . b ) . c = a . b . c

Distributive properties

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

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

Identity Properties

0 + a = a + 0 = a

a . 1 = a . a = a

additive inverse Properties

a + (- a ) = - a +a = 0

Multiplicative inverse properties

a.1a = 1a .a=1 if b ≠ 0

Multiplication by zero

a . 0 = 0

Division properties

0a = 0 aa = 1 if a ≠ 0

Rules of signs

a(-b ) = - (ab) , (-a)b = - (ab) , ( -a ) ( -b ) = ab , - ( -a ) = a , a-b = -ab = - ab , -a-b = ab

Exponents

an = a.a.a…….a n factors , a0 = 1 if a ≠ 0 , a-n = 1an if a ≠ 0

Laws of exponents

anam= am+n , (am)n = amn , abn=anbn , aman=am-n=1an-mif a ≠0 , (ab)n= anbn , if b ≠0

Square roots

a2 =a

Geometry Review

Pythagorean Theorem

c2= a2+b2

Area = πr2

Circumference

= 2πr = πd

Area = 12bh

Geometry Formulas

Area = LW

Perimeter = 2L + 2W

Volume= πr2h

Surface area=

=πr2h+2πrh

Volume= 43πr3

Surface area=4πr2

Volume = LWH

Surface area=

2LW+ 2LH+2WH

Polynomials

Special Products

Difference of two squares

( x – a )( x + a ) = x2- a2

Squares of binomials or perfect squares

( x+a )2= x2+2ax+ a2

( x-a )2= x2-2ax+ a2

Cubes of binomials or perfect Cubes

( x+a )3= x3+3ax3+ 3a2x+a3

( x-a )3= x3-3ax3+ 3a2x+a3

Differences of two cubes

(x-a)(x2+ax+a2)=x3-a3...