real numbers algebra

(N) natural numbers

numbers used for counting (1,2,3,4,5,6,7,8)

(W) whole numbers

natural numbers + 0 (0,1,2,3,4,5,6,7,8,9)

(I) integers

whole numbers + negative numbers (-3,-2,-1,0,1,2,3)

(Q) rational numbers

integers and fractions

(Q’) irrational numbers

non rational numbers, they are non repeating

commutative property of addition

if a and b are real numbers, then a+b = b+a

commutative property of multiplication

if a and b are real numbers, then a x b = b x a

associative property of addition

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

associative property of multiplication

(a x b) x c = a (b x c)

subtraction property

subtracting a number is the same as adding it’s opposite ( a - b = a + (-b) )

distributive property

if a,b,c are real numbers then ( a x (b+c) = ab + ac )

and by the commutative property of multiplication ( (b+c) x a = ba +ca = ab +ac)

identity property of addition

for any real number a: a + 0 = a and 0 + a = a

0 is the additive identity

identity property of multiplication

for any real number a: a x 1 = a and 1 x a = a

1 is the multiplicative identity

inverse property of addition

a + (-a) = 0

for any real number a, -a is the additive inverse of a.

inverse property of multiplication

for any real number a, a ≠ 0, a x 1/a = 1

1/a is the multiplicative inverse of a

division by 0

zero divided by any real number, except itself is zero. division by zero is undefined.

product rule of exponents

a^m x a^n = a^m+n

negative rule of exponents

a^-n = 1/a^n

1/a^-n = a^n

power of the product rule of exponents

(ab)^n = a^n b^n

power of the quotient rule of exponents

(a/b)^n = a^n/b^n

(a/b)^-n = a-n/b-n = b^n/a^n