Properties of Numbers Flashcards
(35 cards)
What are integers?
Which are integers? 3, 5.5, -9, 0, 2.3
Integers are numbers without fractions or decimals.
* Can be positive or negative
* Zero is an integer
* All integers are even (2n) or odd (2n +1). Even defined as being divisible by 2 with no remainder.
3, -9, 0
What are whole numbers?
Which are whole numbers? -4, 1, 0, -52, 16
Whole numbers are non-negative integers.
* Thus, positive integers and zero comprise the set of whole numbers.
1, 0, 16
Is zero negative or positive?
Is zero even or odd?
Zero is neither negative nor positive.
Zero is even.
What are some of the key properties of Zero?
- Zero, divided by any number other than zero, is zero
- Anything divided by zero is undefined
- Zero is the only number that’s neither positive nor negative
- Zero is an even number
- Zero is the only number that is equal to its opposite (e.g., 0 = -0)
-
All numbers are factors of zero, and zero is a multiple of all numbers
(e.g., 3x0 = 0; 0x5 = 0) - Zero is not a factor of any number but itself
There are others, but they are common sense (i.e., anything x 0 = 0)
What are some of the key properties of One?
- One is a factor of all numbers, and all numbers are multiples of 1
- One is the only number with only 1 factor
- One is not a prime factor
There are others, but they are common sense (i.e., n x 1 = n)
How do we define even and odd numbers?
Even: 2n, where n is an integer
Odd: 2n + 1
When adding/subtracting two numbers, what’s a trick to determine if the sum/difference is even or odd?
When adding/subtracting two numbers:
* If both are odd or both are even, the sum/difference is even.
* If not, the sum/difference is odd.
Sums and Differences That Yield Even and Odd Numbers:
Even
* (even) +/- (even) = (even)
* (odd) +/- (odd) = (even)
Odd
* (even) +/- (odd) = (odd)
* (odd) +/- (even) = (odd)
Products That Yield Even and Odd Numbers:
Even
* (even) x (even) = (even)
* (even) x (odd) = (even)
Odd
* (odd) x (odd) = (odd)
Even + Odd = ?
Even + Odd = (Odd)
Odd + Odd = ?
Odd + Odd = Even
Even x Odd = ?
Even x Odd = Even
Odd x Odd = ?
Odd x Odd = Odd
Odd + Odd = ?
Odd + Odd = Even
Quotients That Yield Even and Odd Numbers:
(even) / (even) = (even) or (odd)
(even) / (odd) = (even)
(odd) / (odd) = (odd)
12/6 = 2 ; 12/4 = 3
What is a “signed number?”
A signed number refers to a positive or negative number.
What is an easy way to add numbers with different signs?
e.g., -9 + 3 = ?
Subtract the absolute value of the smaller number from the larger number, then keep the sign from the larger number.
e.g., -9 + 3 = ?
|9| - |3| = |6| –> -6
If we subtract two numbers with opposing signs, how do we approach?
e.g., -7 - 8 = ?
Subtracting is adding the opposite.
We can switch the sign, then use our basic addition rules.
e.g., -7 - 8 = (-7) + (-8) = -15
Two rules for even and odd number of exponents (as they related to positive and negative)
Rule 1: When a nonzero base is raised to an even exponent, the result will be even.
e.g., 3^2 = 9 ; (-2)^4 = 16 ; (-1/2)^6 = (1/64)
Rule 2: When a nonzero base is raised to an odd exponent…
if the base is positive, it will be positive.
If the base is negative, the result will be negative.
e.g., 3^3 = 27 ; (-2)^5 = -32 ; (-1/2)^3 = (-1/8)
What is a factor/divisor?
A factor (or divisor) is a number that divides evenly into another number.
If y is a factor of x, and x is positive:
* The smallest factor of x will be 1, the largest will be x
* y will fall between 1 and x
* For any positive integers x and y, y is only a factor of x if x/y is an integer.
What is a multiple?
A multiple of a number is the product of that number with any integer.
e.g., 2x0 = 0, 2x1 = 2, 2x2 = 4, etc., so 0, 1, 2, 4 are multiples of 2
For two quantities x and y, x is a multiple of y only if x = ny, where n is an integer.
Or: If y=/=0, x is a multiple of y only if x/y is an integer.
Any integer is both a factor and multiple of itself.
e.g., 5 is factor of 5 since 5 divides into 5. 5 is a multiple of 5 since 5x1 = 5
What is a prime number?
What is a composite number?
Prime numbers are any numbers greater than 1 with only two factors: 1 and itself.
Composite numbers are non-prime numbers
2 is the only even prime number
How do we find a number’s total number of factors?
e.g., 120
Step 1.
Find the prime factorization of the number.
e.g., 120 = 2^3 x 3^1 x 5^1
Step 2.
Add 1 to each of the exponents
e.g., (3+1)(1+1)(1+1)
Step 3.
Multiply these numbers, and you have the number of factors.
e.g., (4)(2)(2) = 16, so 120 has 16 factors
If a number is raised to a positive integer exponent, does its number of unique primes increase or decrease?
The number of unique primes does not change.
e.g., 18 = 2^1 x 3^2 … 2 unique primes
Let’s cube 18, giving us 5,832.
5,832 = 2^3 x 3^6 … 2 unique primes
If some number x has y unique prime factors, x^n (where n is a positive integer) will have the same y of unique prime factors.
