Real Numbers
1. Add Numbers2. Subtract Numbers
3. Multiply / Divide Numbers
4. Exponents
5. Simplify Fractions
6. Add/Subtract Fractions Of Same Denominator
7. Add/Subtract Fractions Of Different Denominators
8. Multiply Fractions
9. Divide Fractions
10. Order of Operations
11. Roster method to specify a set of numbers
12. Set-Builder method to specify a set of numbers
13. Intervals
14. Properties of Real Numbers
15. Using the properties to simplify algebraic expressions
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1. Add Numbers
Sum of two negative numbers is a negative number whose absolute value is the sum of the absolute values.
To add two negative numbers:
1. Add the absolute values.
2. Write the minus sign in front.
Sum of two numbers of opposite signs is a number:
♦ whose absolute value is the difference of the absolute values,
♦ is positive if the positive addend has larger absolute value,
♦ is negative if the negative addend has larger absolute value.
To add two numbers of opposite signs:
1. Subtract the absolute values.
2. If the negative number has larger absolute value then write the minus sign in front.
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2. Subtract Numbers
Subtract a number is to add its opposite.
To subtract two numbers:
1. Change the subtraction to addition.
2. Change the sign of the second number.
3. Add.
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3. Multiply / Divide Numbers
♦ Product/Quotient of a positive number and a negative number is a negative number.
♦ Product/Quotient of two negative numbers is a positive number.
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4. Exponents
An exponential expression is a notation of the form bn, where b is called the base, and n is called the exponent of b.
This expression indicates that b is multiplied with itself n times.
Examples. b1 = b b2 = b · b b3 = b · b · b b4 = b · b · b · b
Examples. 21 = 2 22 = 2 · 2 = 4 23 = 2 · 2 · 2 = 8 24 = 2 · 2 · 2 · 2 = 16
25 = 2 · 2 · 2 · 2 · 2 = 32 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
Examples. ( − 2 )1 = − 2 ( − 2 )2 = ( − 2 ) · ( − 2 ) = 4 ( − 2 )3 = ( − 2 ) · ( − 2 ) · ( − 2 ) = − 8
( − 2 )4 = ( − 2 ) · ( − 2 ) · ( − 2 ) · ( − 2 ) = 16
Examples. ( − 3 )1 = − 3 ( − 3 )2 = 9 ( − 3 )3 = − 27 ( − 3 )4 = 81
Examples. Perfect squares are
12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100
Examples. Perfect cubes are
13 = 1 23 = 8 33 = 27 43 = 64 53 = 125
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5. Simplify Fractions
1. Find the common factor of the numerator and the denominator.2. Divide the numerator by this common factor.
3. Divide the denominator by this common factor.
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6. Add/Subtract Fractions Of Same Denominator
1. Add/subtract numerators.2. Keep the same denominator.
3. Simplify, if needed.
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7. Add/Subtract Fractions Of Different Denominators
1. Find the new denominator: multiply the denominators.2. Find the new numerator: cross-multiply the denominators to the numerators, then add/subtract.
3. Simplify, if needed.
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8. Multiply Fractions
1. If a numerator and a denominator have a common factor then divide them both by this factor.2. Find the new numerator: multiply the numerators.
3. Find the new denominator: multiply the denominators.
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9. Divide Fractions
1. Change the division to multiplication.2. Change the divisor (on the right) to its reciprocal.
3. Multiply.
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10. Order of Operations
To evaluate the value of an algebraic expression without grouping symbols, we carry out the operations in the following order
1. Exponents.
2. Multiplications and divisions from left to right.
3. Additions and subtractions from left to right.
Example. Evaluate 2 · 32 − 30 ÷ 2 · 5 + 10 .
Answer. = 2 · 9 − 30 ÷ 2 · 5 + 10
= 18 − 15 · 5 + 10
= 18 − 3 + 10
= 15 + 10
= 25
To evaluate the value of an algebraic expression with grouping symbols, we reduce the expression in each grouping symbol to a single number using the above order. If the grouping symbols are nested, carry out the operations from the inner grouping symbols first.
Example. Evaluate ( 2 · 3 )2 − [ 30 ÷ ( 2 · 5 ) + 10 ] .
Answer. = ( 6 )2 − [ 30 ÷ ( 10 ) + 10 ]
= 36 − [ 3 + 10 ]
= 36 − [ 13 ]
= 23
Note that − 24 = − 16 whereas ( − 2 )4 = 16.
and − 23 = − 8 ( − 2 )3 = − 8.
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11. Roster method to specify a set of numbers
♦ The numbers are listed between a pair of braces.
♦ The numbers are separated by commas.
Example. Use roster method to specify the set of all odd integers from 3 to 11.
Answer. { 3 , 5 , 7 , 9 , 11 }
Example. Use roster method to specify the set of all positive integers from 3 to 31,
which are multiples of 5.
Answer. { 5 , 10 , 15 , 20 , 25 , 30 }
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12. Set-Builder method to specify a set of numbers
1. A pair of braces which is translated to "this is a set of".
2. Inside the braces is a letter, normally x, which is translated to "real numbers named x"
3. Follow the letter is a vertical bar, | , which is translated to "with the following properties"
4. Follow the vertical bar is a list of properties that are common and unique to all numbers in the set.
Example. Use set-builder method to specify the set of all real numbers between 3 and 4.8, inclusively.
Answer. { x | 3 ≤ x ≤ 4.8 }
Example. Use set-builder method to specify the set of all real numbers greater than 2 and less than or equal to 7.
Answer. { x | 2 < x ≤ 7 }
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13. Intervals
♦ An interval is a set of all real numbers between two given numbers (or infinity) ∞, called endpoints.
♦ Interval notation is normally used to specify an interval
1. Smaller endpoint is written on the left, larger endpoint on the right, separated by a comma.
2. On the left side of the left endpoint
Use a square bracket [ to indicate that the endpoint is included in the interval,
Use a parenthesis ( to indicate that the endpoint is included in the interval.
3. On the right side of the right endpoint
Use a square bracket ] to indicate that the endpoint is included in the interval,
Use a parenthesis ) to indicate that the endpoint is included in the interval.
Example. Use interval notation to specify the set { x | 2 < x < 7 }.
Answer. ( 2 , 7 )
Example. Use interval notation to specify the set { x | 2 ≤ x < 7 }.
Answer. [ 2 , 7 )
Example. Use interval notation to specify the set { x | 2 < x ≤ 7 }.
Answer. ( 2 , 7 ]
Example. Use interval notation to specify the set { x | 2 ≤ x ≤ 7 }.
Answer. [ 2 , 7 ]
Example. Use interval notation to specify the set { x | x < 7 }.
Answer. ( - ∞ , 7 )
Example. Use interval notation to specify the set { x | x ≤ 7 }.
Answer. ( - ∞ , 7 ]
Example. Use interval notation to specify the set { x | x > 2 }.
Answer. ( 2 , ∞ )
Example. Use interval notation to specify the set { x | x ≥ 2 }.
Answer. [ 2 , ∞ )
Example. Use interval notation to specify the set of all real numbers.
Answer. ( - ∞ , ∞ )
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14. Properties of Real Numbers
1. Additive commutative: a + b = b + a
Example. 2 x + 5 = 5 + 2 x
Example. 2 x - 5 = - 5 + 2 x
2. Additive associate: ( a + b ) + c = a + ( b + c )
Example. ( 2 x + 5 ) + 3 = 2 x + ( 5 + 3 )
3. Zero is additive identity: a + 0 = 0 + a = a
Example. 2 x + 0 = 2 x
Example. 0 + 2 x = 2 x
4. Additive inverse of a is - a: a + ( - a ) = 0
Example. 2 x + ( - 2 x ) = 0
Example. - 2 x + 2 x = 0
Example. - ( - 2 x ) = 2 x
5. Multiplicative commutative: a · b = b · a
Example. 2 x · 5 = 5 · 2 x
6. Multiplicative associate: ( a · b ) · c = a · ( b · c )
Example. 2 · ( 5 · x ) = ( 2 · 5 ) · x
7. One is multiplicative identity: 1 · a = a · 1 = a
Example. 2 x · 1 = 2 x
Example. 1 · 2 x = 2 x
8. Multiplicative inverse of a is 1⁄a: a · 1⁄a = 1
Example. 2 x · 1⁄2 x = 1
Example. 1⁄2 x · 2 x = 1
Example. 1⁄1⁄2 x = 2 x
9. Distributive Property: a · ( b + c ) = a · b + a · c
Example. 3 ( 2 x + 5 ) = 6 x + 15
Example. 3 ( 2 x - 5 ) = 6 x - 15
Example. - 3 ( 2 x - 5 ) = - 6 x + 15
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15. Using the properties to simplify algebraic expressions
Example. Simplify 3 ( 2 x + 5 ) + 5 ( 2 x - 3 )
Answer. = ( 6 x + 15 ) + ( 8 x - 20 ) distributive
= 6 x + 15 + 10 x - 15 additive associative
= 6 x + 10 x + 15 - 15 additive commutative
= 6 x + 10 x + 0 additive inverse
= ( 6 + 10 ) x + 0 (reversed) distributive
= 16 x + 0
= 16 x additive identity