THEOREMS ON REAL NUMBERS

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

I. PROPERTIES OF REAL NUMBERSCertain Basic Theorems

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 1. Cancellation Law for Addition. If a+c=b+c then a=bTheorem 2. Cancellation Law for Multiplication. If ac=bc, c≠0 then a=b.

I. PROPERTIES OF REAL NUMBERSCertain Basic Theorems

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 3. If a=b then -a=-b (If two numbers are equal then their negatives are also equal)Theorem 4. For any real number a, ax0=0. (The product of any real number and zero is equal to zero)

I. PROPERTIES OF REAL NUMBERSCertain Basic Theorems

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 5. ab=0 if and only if a=0 or b=0.(Whenever the product of two numbers is zero, one or both of the numbers is zero)Theorem 6. –(-a) = a. (The negative of negative of a is equal to a.)

I. PROPERTIES OF REAL NUMBERSCertain Basic Theorems

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 7. For any non-zero real number a, 1/(1/a) =a.(The reciprocal of the reciprocal of a non-zero number is equal to the non-zero number a)

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Definition. Subtraction. Subtracting b from a denoted by a-b, is equal to adding the negative of b to a.That is, a-b=a+(-b)

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 8. a(b-c)=ab-ac (Distributive axiom).Theorem 9. -(a+b)=-a-b-(a-b)=-a+b

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Definition. Division. The quotient of a by b, written as a/b, is equivalent to multiplying a by the reciprocal of b, b≠0.That is, a/b=a (1/b), b≠0.

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 10. For any non-zero real number a, a/1 =a.For any real number b, b/1 =b.Theorem 11.

, 0a b a b cc c c

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 12.

, 0ac a cbc b

Theorem 13. , , 0a c ac b d

b d bd

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 14.

Theorem 15. , 0, 0a c ad bc b d

b d bd

, 0, 0a c iffad bc b db d

I. PROPERTIES OF REAL NUMBERSTheorems involving Subtraction and Division

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

Theorem 16.

Theorem 17. for any real numbers a and b, b≠0,

, , , 0a c a d b c db d b c

a a a aandb b b b

UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra

1. College Algebra and Trigonometry by Leithold

2. Mathematics Modules. Summer Bridge Program, Summer 200 by UP-LRC

3. Workbook for PSI for College Algebra by UP-LRC

References