MORE DEFINITIONS AND BASIC PROPERTIES OF NUMBERS

 

SIGNED NUMBERS

 

Absolute Value:  The absolute value of a number is the positive value of the number.  Technically, it is the distance from zero on the number line.  We write |x| to symbolize the absolute value of x.

Example:  |5| = +5

Example:  |-5| = +5

 

Positive Number:  A number greater than 0.

 

Negative Number:  A number less than 0.

 

Note:  0 is neither positive nor negative.

Note:  We refer to positive and negative numbers as "signed numbers."

 

Operations with Positive and Negative Numbers

 

Addition of Signed Numbers:  If two numbers have the same sign, add the absolute value of the numbers.  The answer will have the same sign as the two numbers.

Example:  (+8) + (+3) = +11

Example:  (-8) + (-3) = -11

If two numbers have different signs, subtract the absolute value of the smaller number from the absolute value of the larger number.  The answer will have the sign of the larger number.

Example:  (+8) + (-3) = +5

Example:  (-8) + (+3) = -5

 

Subtraction of Signed Numbers:  Rather than subtracting, change the sign of the number that is being subtracted and follow the rules for addition.

Example:  (+9) - (+7) = (+9) + (-7) = +2

Example:  (+9) - (-7) = (+9) + (+7) = +16

Example:  (-9) - (+7) = (-9) + (-7) = -16

Example:  (-9) - (-7) = (-9) + (+7) = -2

 

Multiplication of Signed Numbers:  If the signs are the same, the answer will be positive.  If the signs are different, the answer will be negative.

Example:  (+4) x (+6) = +24

Example:  (+4) x (-6) = -24

Example:  (-4) x (+6) = -24

Example:  (-4) x (-6) = +24

 

Division of Signed Numbers:  The rules for division are the same as the rules for multiplication.  If the signs are the same, the answer will be positive.  If the signs are different, the answer will be negative.

Example:  (+24)  (+8) = + 3

Example:  (+24)  (-8) = - 3

Example:  (-24)  (+8) = - 3

Example:  (-24)  (-8) = + 3

 

 

ORDER OF NUMBERS

 

If a is less than b we write a < b.

If a is less than or equal to b we write a  b.

If a is greater than b we write a > b.

If a is greater than or equal to b we write a  b.

 

Example:  3 < 5

Example:  If x is an integer and x   5, then x could be 5 or any integer greater than 5.

 

Solving inequalities is just like solving equalities with one important exception:  When we multiply or divide both sides of an inequality we change the direction of the inequality sign.  For example, we know that

 

3 > 2. 

 

However, multiplying  3 by -1 and 2 by -1 "flips" the inequality sign:

 

-3 < -2

 

Example:  What values of x satisfy the inequality x - 3 > 6?

Add 3 to both sides of the equation to get x -3 + 3 > 6 + 3.

Therefore, x > 9.

 

Example:  What values of x satisfy the inequality -4x  < 36?

Divide both sides of the equation by -4.  Note, however, that the direction of the inequality sign changes:

Therefore, x > -9.

 

 

PRIME AND COMPOSITE NUMBERS

 

Prime number:  A positive integer greater than one that has no divisors other than itself and one. 

 

Examples:  2, 3, 5, 7, 11, 13, 17, 19, …

 

Composite number:  A positive integer that is not prime.

 

Note:  1 is neither prime nor composite.

 

 

EVEN AND ODD NUMBERS

 

Even number:  An integer that is evenly divisible by 2.  Algebraically, we can represent an even number as 2n where n is an integer.  Note that 0 is even since 0 ¸ 2 = 0 with remainder 0.

 

Odd number:  An integer that is not even.  Algebraically we can represent an odd number as 2n + 1 where n is an integer. 

 

Fact

 

Example

Algebraic Approach

even + even = even

4 + 6 = 10

2x + 2y = 2(x+y)

even + odd = odd

4 + 7 = 11

2x + (2y+1) = 2 (x+y) +1

odd + odd = even

5 + 7 = 12

(2x+1)+(2y+1) = 2(x+y+1)

 

 

 

even x even = even

4 x 6 = 24

2x·2y = 4xy = 2(2xy)

even x odd = even

4 x 7 = 28

(2x+1) · 2y=4xy+2y=2(2xy+y)

odd x odd = odd

5 x 7 = 35

(2x+1)(2y+1) = 4xy+2x+2y+1

= 2(2xy+x+y)+1

 

GREATEST COMMON FACTOR (GCF)

 

The greatest common factor of two numbers is the largest factor that they have in common.

 

Example:  find the GCF of 24 and 36.

Factors of 24:  1, 2, 3, 4, 6, 8, 12, 24

Factors of 36:  1, 2, 3, 4, 6, 9, 12, 18, 36

The largest factor common to both 24 and 36 is 12.

 

We can also compute the prime factorization of both numbers and find the prime factors that are common to both numbers.

 

24 = 2x2x2x3

36 = 2x2x3x3

 

The factors that occur in both prime factorizations are 2, 2, and 3.

Therefore, the GCF = 2x2x3 = 12.

 

 

LEAST COMMON MULTIPLE (LCM)

 

The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

 

Example:  Find the LCM of 18 and 24.

Multiples of 18:  18, 36, 72, 90, 108, 126, 144, 162 …

Multiples of 24:  24, 48, 96, 120, 144, 168 …

The smallest multiple of both 18 and 24 is 144.

 

Shortcut:  We don't have to write all the multiples of both numbers.  Just look at the multiples of one number, say 24.  Then test numbers to find the first one that is divisible by 18.

Example:  Let x represent a positive even number and y represent a positive odd number.  Which of the following are odd?

 

I.

II.

III.  

 

A.  I only

B.  II only

C.  III only

D.  II and III only

E.  I, II and III

 

Answer:  C

Solution:  Plug in numbers.  Let x = 4 and y = 3. 

I.  = 4·4·4 = 64.  Note, however, that any number multiplied by an even number is even, so regardless of the exponent, if the base is even, that base raised to any power is even.

II.  = 16 which is even.  Again, however, an even number raised to any power is even.

III.  = 81 which is odd.  We know the product of two odd numbers will also be odd, and if that product is again multiplied by an odd number the new product will be odd.  Therefore,  is odd, as long as x > 0.

Example:  Eric and Tamara both work.  Eric has every sixth day off and Tamara has every eighth day off.  How often do they have a day off together?

 

Answer:  Every 24 days.

Assume they both start work on the first day of a month.  Then Eric has the 6th, 12th, 18th, 24th and 30th day of the month off.   Tammy has the 8th, 16th, 24th and (32nd??) day of the month off.  Since they are both off work on the 24th, every 24 days they have a day off together.  In this problem we were looking for the lowest common multiple of 6 and 8.

 

Example:  Lisa has a number of plants she plans to plant.  She prefers to have the same number of plants in each row in her garden.  If she plants the plants in 2 rows, she will have one plant left over.  If she plants the plants in 3 rows, she will again have one left over.  If she plants them in 4, 5, 6, or even 10 or 12 rows, she will always have one left over.  What is the fewest number of plants she could have?

 

If we remove one plant, the remaining plants can be planted in rows of 2, 3, 4, 6, 10, or 12, and the number of plants must be a multiple of each of these numbers.  We want to find the smallest number that is divisible by all of these numbers.  That is, we want the LCM of these numbers.  Then the LCM + 1 will have a remainder of 1 when it is divided by any of these numbers.

 

Luckily, we do not have to look at the multiples of all the numbers.  Since the ACT and the SAT do not require large amounts of calculation there must be a shortcut:

 

Since 12 is a multiple of 2, 4, and 6, any multiple of 12 will be a multiple of these numbers.  Similarly, since 10 is a multiple of 5, any multiple of 10 will also be a multiple of 5.  Therefore, we merely need to find the LCM of 10 and 12.  The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, … .  Since 60 is the first multiple of 10 that is divisible by 12, 60 is the LCM.  Therefore, Lisa has 60 + 1 = 61 plants.