# Properties of Numbers and Operations

Number sentence

Primary Conjecture

Property & conditions

True/False Statements

n + 0 = n When you add zero to a number you get that same number.

For every real number n + 0 = n

• 8 + 0 = 8
• 7 + 3 = 11
• 367 + 0 = 0
• 369 + 0 = 369
• 30 + 0 = 300
• 2 zebras + 0 zebras = 2 zebras
• 4z + 0 z = 4 z
• x + 0 = 0
• x + 0 = x
0 + n = n When you add a number to zero you get that same number.
• 0 + 8 = 8
• 4 + 6 = 11
• 0 + 763 = 0
• 763 + 0 = 763
• 0 + 30 = 300
• 0 + 2 turtles = 2 turtles
• 0T + 3T = 3 T
• 0 + x = 0
• X + 0 = 0
• x + 0 = x
n - 0 = n When you subract zero from a number you get that same number.

Identity property of subtraction:

For every real number n - 0 = n

• 8 - 0 = 8
• 7 - 3 = 10
• 367 - 0 = 0
• 369 - 0 = 369
• 30 - 0 = 3
• 2 zebras - 0 zebras = 2 zebras
• 4z - 0 z = 4 z
• x - 0 = 0
• x - 0 = x
n - n = n When you subract a number from itself you get zero.

For every real number n, there is a real number -n such that n + (-n) = 0

• 3 - 3 = 0
• 3 - 2 = 0
• 8 - 8 = 0
• 2 trees - 2 trees = 0
• 2t - 2t = 0
• b - b = 0
• 9 - b = 0
• 29 - n = 0
• n - b = 0

## Multiplication and Division properties

Number sentence

Primary Conjecture

Property & conditions

True/False Statements

n * 1 = a When you multiply a number times 1 you get that same number.

Identity property of multiplication:

For all real numbers a, a * 1 = a

3 * 1 = 3
1 * n = n When you multiply 1 times a number, you get that number.   1 * 4 = 4
n / 1 = n When you divide a number by 1, you get that number.   5 / 1 = 5
n / n = 1, n ≠ 0 When you divide a number by itself you get one, except when the number is zero.   6 / 6 = 1
n * 1/n = 1

Inverse property of multiplication:

For every real number n, n ≠ 0, there is a real number 1/n such that n * 1/n = 1

3 * 1/3 = 1

## Multiplication and Division with zero

Number sentence Primary Conjecture Property & conditions True/False Statements
n * 0 = 0
When you multiply a number times zero, you get zero. Property of zero 2 * 0 = 0
0 * n = 0 When you multiply a zero times a number, you get zero.   0 * 3 = 0

0 / n = 0,

a ≠ 0

When you divide zero by any number except zero you get zero.   0 / 0 = 0

## Commutative property for addition and multiplication

Number sentence Primary Conjecture Property & conditions True/False Statements
a + b = b + a When you add two numbers, you can change the order of the numbers you add, and you wlll get the same answer.

the order in which a pair of addends is added does not affect the sum.

For all real numbers a and b,
a + b = b + a

4 + 3 = 3 + 4

1 + (2 + 3) =
1 + (3 + 2)

a * b = b * a When you multiply two numbers, you can change the order of the numbers you multiply, and you will get the same answer.

Commutative property of multiplication:

the order in which a pair of factors is multiplied does not affect the product.

For all real numbers a and b,
a * b = b * a

2 * 3 = 3 * 2

2 (3 * 4) = 2(4 * 3)

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

Distributive property of multiplication over addition:

For all real numbers a, b, c,
a * (b + c) = (a * b) + (a * b)

2(3 * 4) =
(2 * 3) + (2 * 4)