AoPS Ch 1 Properties of Arithmetic

Question 0

Addition is commutative. What does this mean?

Answer 0

a + b = b + a

Question 1

Order of operations: In what order do you perform the operations in an expression?

Answer 1

1. Evaluate expressions inside parentheses first.
2. Compute powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Question 2

What is another way to write 3 + 7 using the commutative property of addition?

Answer 2

7 + 3

Question 3

Addition is associative. What does this mean?

Answer 3

(a + b) + c = a + (b + c)

Question 4

What is another way to write (2 + 5) + 3 using the associative property of addition?

Answer 4

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.

Question 5

Compute (35 + 13) + 27

Answer 5

Using the associative property:

35 + (13 + 27) = 35 + 40 = 75

Question 6

Multiplication is commutative: What does this mean?

Answer 6

ab = ba

Question 7

Multiplication is associative. What does this mean?

Answer 7

(ab)c = a(bc)

Question 8

According to the commutative property, what does 7x35 equal?

Answer 8

7 x 35 = 35 x 7

Question 9

Using the associative property, compute 25 x (12 x 8) in your head.

Answer 9

25 x (12 x 8) = (25 x 12) x 8 = 300 x 8 = 2400

Question 10

Distributive Property: Multiplication distributes over addition. What does this mean?

Answer 10

a(b + c) = ab + ac

and

(b + c)a = ba + ca

Question 11

Using the distributive property, what is 5 x 17?

Answer 11

5 x 17 = 5 x (10 + 7) = 5 x 10 + 5 x 7 = 50 + 35 = 85

Question 12

Using the distributive property, compute (7 + 9) x 13.

Answer 12

(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

Question 13

What is the Negation Property?

Answer 13

-a + a = 0

Question 14

What is the Reciprocal Property?

Answer 14

(1/a) x a = 1

Question 15

What is the definition of SUBTRACTION?

Answer 15

a - b = a + (-b)

Subtraction is addition of the negation.

Question 16

What is DIVISION?

Answer 16

(a/b) = a x (1/b)

Division is the same as multiplication by the reciprocal.

Question 17

Negation of a negation says the -(-a) = ?

Answer 17

-(-a) = a

Question 18

Negation of a sum says that -(a + b) = ?

Answer 18

-(a + b) = (-a) + (-b)

Question 19

Multiplying by -1 means that (-1)a = ?

Answer 19

(-1)a = -a

Question 20

Multiplying by a Negation means that (-a)b = ?

Answer 20

(-a)b = -(ab)

and

a(-b) = -(ab)

Question 21

Negation times Negation means that (-a)(-b) = ?

Answer 21

(-a)(-b) = ab

Question 22

Compute (-3)(-2) =

Answer 22

(-3)(-2) = 6

Question 23

Compute (-3)(2) =

Answer 23

(-3)(2) = -6

Question 24

Compute (3)(-2) = ?

Answer 24

(3)(-2) = -6

Question 25

Compute (-1)(7) =

Answer 25

(-1)(7) = -7

Question 26

Subtracting from 0: 0 - a =

Answer 26

-a

Question 27

Self subtraction: a - a =

Answer 27

0

Question 28

Subtracting 0: a - 0 =

Answer 28

a

Question 29

Subtraction of negation: a - (-b) =

Answer 29

a + b

Question 30

Subtraction from negation: -a - b =

Answer 30

-(a+b)

Question 31

Negation of subtraction: -(a - b) =

Answer 31

b - a

Question 32

Multiplication distributes over subtraction: a(b - c) =

Answer 32

ab - ac

Question 33

Multiplication distributes over subtraction: (b - c)a =

Answer 33

ba - ca

Question 34

Reciprocal of reciprocal: If a is nonzero, then the reciprocal of (1/a) is ?

Answer 34

a

Question 35

Reciprocal of negation: If a is nonzero, then (1/-a) =

Answer 35

-(1/a)

Question 36

Reciprocal of product: If a and b are nonzero, then (1/ab) =

Answer 36

(1/a) x (1/b)

Question 37

Dividing into zero: If a is nonzero, then 0/a =

Answer 37

0

Question 38

Self division: If a is nonzero, then a/a =

Answer 38

1

Question 39

Dividing by 1: a/1 =

Answer 39

a

Question 40

Dividing into 1: If x is nonzero, then 1 divided by x is ?

Answer 40

1/x

Question 41

Dividing by reciprocal: If b is nonzero, then a divided by (1/b) is

Answer 41

ab

Question 42

Dividing into negation: If b is nonzero, then (-a)/b =

Answer 42

-(a/b)

Question 43

Dividing by negation: If b is nonzero, then a/(-b) =

Answer 43

-(a/b)

Question 44

Negation divided by negation: If b is nonzero, then (-a)/(-b) =

Answer 44

a/b

Question 45

Cancel common factor: If a and c are nonzero, then (ab)/(ac) =

Answer 45

b/c

Question 46

Division by a number distributes over addition: If c is nonzero, then (a + b)/c =

Answer 46

a/c + b/c

Question 47

Division by a number distributes over subtraction: If c is nonzero, then (a - b)/c =

Answer 47

a/c - b/c

Question 48

Adding zero: a + 0 =

Answer 48

a

Question 49

Multiplying by zero: 0 x a = 0a =

Answer 49

0

Question 50

Subtraction is neither commutative nor associative. What does this mean?

Answer 50

a - b does not necessarily equal b - a, and | a - (b - c) does not necessarily equal (a - b) - c

Question 51

You can't divide by

Answer 51

Zero. Division by zero is undefined. Before dividing by a number, be sure that the number is nonzero.

Question 52

Division is neither commutative nor associative. What does this mean?

Answer 52

a/b and b/a are not necessarily equal, and | a/(b/c) and (a/b)/c are not necessarily equal.

Question 53

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?

Answer 53

(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 ```