Download Lecture notes for Section R.3

Int. Alg. Notes
Section R.3
Page 1 of 5
Section R.3: Operations on Signed Numbers
Big Idea: Since your instructor considers algebra to be the study of how to perform multi-step arithmetic
calculations more efficiently, he thought it would be a good idea to review how to perform basic, one-step
arithmetic calculations.
Big Skill: You should be able to perform arithmetic operations involving signed numbers and rational numbers
(i.e., fractions), and be able to use the distributive property of real numbers.
What is algebra?
1. It is the study of how to perform multi-step arithmetic calculations more efficiently, and
2. It is the study of how to find the correct number to put into a multi-step calculation to get a desired
answer.
Example:
1. Calculating the total price of a purchase after sales tax.
2. Calculating the price before tax is added when you know what total price you can afford.
Review of how to perform single-step arithmetic calculations on signed numbers and fractions:
Adding signed numbers:
1. If both numbers are positive, then just add the numbers.
2. If both numbers are negative, then add the absolute values of each number, and take the negative of that
answer.
3. If one number is positive, and one number is negative, then subtract the smaller absolute value from the
larger absolute value; the sign of the answer is the sign of the larger absolute value.
Subtracting signed numbers:
Add the negative of the second number to the first number.
Multiplying or dividing signed numbers:
Multiply or divide the absolute values of each number; the answer is negative only if the numbers have opposite
sign.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section R.3
Page 2 of 5
Practice:
a. -11 + 3 =
b. 9.3 + (-6.4) =
c. -15 + (-88) =
d. -7.3 – (-4.2) =
e. -3 – 8 =
f. -68 =
g. (-1.9)(-2.7) =
h. (-10)  2 =
i.
1

5
Multiplying rational numbers:
Multiply numerator times numerator and denominator times denominator.
Dividing rational numbers:
Take the reciprocal of the second rational number, then multiply.
Adding or Subtracting rational numbers:
Convert each rational number to its equivalent over a common denominator, then add or subtract the
numerators.
To find the least common denominator:
1. Factor each denominator.
2. Write down the common factor(s), and then copy any additional factors.
3. Multiply those numbers; the product is the least common denominator.
4. Convert each rational number to an equivalent fraction with the LCD as denominator.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section R.3
Page 3 of 5
Practice:
8 15
 
a.
3 4
3 6
b.   
5 7
c.
1
2
2

5
d.
5 11
 
12 12
e.
4 7
 
15 15
f.
8 5
 
15 12

g. 
5 1


18 45
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section R.3
Page 4 of 5
The Distributive Property of Real Numbers:
If a, b, and c are real numbers, then
a  b  c   a  b  a  c
 a  b  c  a  c  b  c
The distributive property is most useful for “removing parentheses,” which can make multi-step calculations
more efficient.
Practice: Use the distributive property to remove the parentheses.
a. 2  5  3 
b. 2  x  3 
c. 3  2 y  1 
d.
 z  4  3 
e.
1
 4 x  10  
2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section R.3
Page 5 of 5
Properties of Real Numbers
1. Additive Inverse Property: For any real number a other than 0, there is another real number
called –a, called the additive inverse, or opposite, of a, having the following property:
a   a    a   a  0
2. Identity Property of Addition: For any real number a, 0  a  a  0  a .
3. Commutative Property of Addition: For any real numbers a and b, a  b  b  a  0 .
4. Commutative Property of Multiplication: For any real numbers a and b, a  b  b  a .
5. Identity Property of Multiplication: For any real number a, a 1  1 a  a .
6. Multiplicative Inverse Property: For any real number a other than 0, there is another real
1
number , called the multiplicative inverse, or reciprocal, of a, having the following property:
a
1 1
a  a 1
a a
7. Multiplication By Zero: For any real number a, the product of a and zero is zero:
a  0  0 a  0
8. Division Properties: For any real nonzero number a,
0
a
a
 0 ,  1 , and
is undefined.
a
0
a
9. Reduction Property: If a, b, and c are real numbers, then
ac a
 , if b  0 and c  0.
bc b
10. Associative Property:
a  b  c    a  b  c  a  b  c
a  b  c    a  b  c  a  b  c
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.