AoPS Ch 1 Properties of Arithmetic Flashcards
Addition is commutative. What does this mean?
a + b = b + a
Order of operations: In what order do you perform the operations in an expression?
- Evaluate expressions inside parentheses first.
- Compute powers.
- Multiply and divide from left to right.
- Add and subtract from left to right.
What is another way to write 3 + 7 using the commutative property of addition?
7 + 3
Addition is associative. What does this mean?
(a + b) + c = a + (b + c)
What is another way to write (2 + 5) + 3 using the associative property of addition?
2 + (5 + 3)
In other words, you can add 5+3 first to get 8 and then add 2 to get 10, instead of 2+5 to get 7 and then 7+3 to get 10.
Compute (35 + 13) + 27
Using the associative property:
35 + (13 + 27) = 35 + 40 = 75
Multiplication is commutative: What does this mean?
ab = ba
Multiplication is associative. What does this mean?
(ab)c = a(bc)
According to the commutative property, what does 7x35 equal?
7 x 35 = 35 x 7
Using the associative property, compute 25 x (12 x 8) in your head.
25 x (12 x 8) = (25 x 12) x 8 = 300 x 8 = 2400
Distributive Property: Multiplication distributes over addition. What does this mean?
a(b + c) = ab + ac
and
(b + c)a = ba + ca
Using the distributive property, what is 5 x 17?
5 x 17 = 5 x (10 + 7) = 5 x 10 + 5 x 7 = 50 + 35 = 85
Using the distributive property, compute (7 + 9) x 13.
(7 + 9) x 13 = (7 x 13) + (9 x 13) = 91 + 117 = 208
or
(7+9) x 13 = 16x13 = 16 x (10+3) = 16x10 + 16x3 = 160+48 = 208
What is the Negation Property?
-a + a = 0
What is the Reciprocal Property?
(1/a) x a = 1
What is the definition of SUBTRACTION?
a - b = a + (-b)
Subtraction is addition of the negation.
What is DIVISION?
(a/b) = a x (1/b)
Division is the same as multiplication by the reciprocal.
Negation of a negation says the -(-a) = ?
-(-a) = a
Negation of a sum says that -(a + b) = ?
-(a + b) = (-a) + (-b)
Multiplying by -1 means that (-1)a = ?
(-1)a = -a
Multiplying by a Negation means that (-a)b = ?
(-a)b = -(ab)
and
a(-b) = -(ab)
Negation times Negation means that (-a)(-b) = ?
(-a)(-b) = ab
Compute (-3)(-2) =
(-3)(-2) = 6
Compute (-3)(2) =
(-3)(2) = -6
Compute (3)(-2) = ?
(3)(-2) = -6
Compute (-1)(7) =
(-1)(7) = -7
Subtracting from 0:
0 - a =
-a
Self subtraction:
a - a =
0
Subtracting 0:
a - 0 =
a
Subtraction of negation:
a - (-b) =
a + b
Subtraction from negation:
-a - b =
-(a+b)
Negation of subtraction:
-(a - b) =
b - a
Multiplication distributes over subtraction:
a(b - c) =
ab - ac
Multiplication distributes over subtraction:
(b - c)a =
ba - ca
Reciprocal of reciprocal:
If a is nonzero, then the reciprocal of (1/a) is ?
a
Reciprocal of negation:
If a is nonzero, then (1/-a) =
-(1/a)
Reciprocal of product:
If a and b are nonzero, then (1/ab) =
(1/a) x (1/b)
Dividing into zero:
If a is nonzero, then 0/a =
0
Self division:
If a is nonzero, then a/a =
1
Dividing by 1:
a/1 =
a
Dividing into 1:
If x is nonzero, then 1 divided by x is ?
1/x
Dividing by reciprocal:
If b is nonzero, then a divided by (1/b) is
ab
Dividing into negation:
If b is nonzero, then (-a)/b =
-(a/b)
Dividing by negation:
If b is nonzero, then a/(-b) =
-(a/b)
Negation divided by negation:
If b is nonzero, then (-a)/(-b) =
a/b
Cancel common factor:
If a and c are nonzero, then (ab)/(ac) =
b/c
Division by a number distributes over addition:
If c is nonzero, then (a + b)/c =
a/c + b/c
Division by a number distributes over subtraction:
If c is nonzero, then (a - b)/c =
a/c - b/c
Adding zero:
a + 0 =
a
Multiplying by zero:
0 x a = 0a =
0
Subtraction is neither commutative nor associative. What does this mean?
a - b does not necessarily equal b - a, and
a - (b - c) does not necessarily equal (a - b) - c
You can’t divide by
Zero.
Division by zero is undefined. Before dividing by a number, be sure that the number is nonzero.
Division is neither commutative nor associative. What does this mean?
a/b and b/a are not necessarily equal, and
a/(b/c) and (a/b)/c are not necessarily equal.
We can use the distributive property to divide a sum by a number, but we can’t use the distributive property to divide a number by a sum. What does this mean?
(a + b)/c = a/c + b/c and (a - b)/c = a/c - b/c
but a/(b + c) is not necessarily equal to a/b + a/c and a(b - c) is not necessarily equal to a/b - a/c
