Important formulas and concepts related to integers for Grade 6

 Here are some important formulas and concepts related to integers for Grade 6:

Basic Operations with Integers

1. Addition of Integers

   • Same sign: Add the numbers and keep the sign.
     • Example: $(+3) + (+5) = +8$
     • Example: $(-4) + (-7) = -11$
   • Different signs: Subtract the smaller number from the larger one and take the sign of the larger number.
     • Example: $(+6) + (-4) = +2$
     • Example: $(-9) + (+3) = -6$

2. Subtraction of Integers

   • Change the subtraction sign to addition and take the opposite sign of the second number. Then follow the rules of addition.
     • Example: $(+8) - (+5) = (+8) + (-5) = +3$
     • Example: $(-6) - (-2) = (-6) + (+2) = -4$

3. Multiplication of Integers

   • Same signs → Positive result
     • Example: $(+4) \times (+3) = +12$
     • Example: $(-5) \times (-2) = +10$
   • Different signs → Negative result
     • Example: $(+6) \times (-2) = -12$
     • Example: $(-7) \times (+3) = -21$

4. Division of Integers

   • Same signs → Positive result
     • Example: $(+12) \div (+4) = +3$
     • Example: $(-15) \div (-5) = +3$
   • Different signs → Negative result
     • Example: $(+20) \div (-4) = -5$
     • Example: $(-18) \div (+6) = -3$

Important Properties of Integers

1. Closure Property

   • Addition and multiplication of two integers always result in an integer.
     • Example: $5 + (-3) = 2$ (integer)
     • Example: $(-4) \times (-3) = 12$ (integer)

2. Commutative Property

   • Addition and multiplication are commutative:
     • $a + b = b + a$
     • $a \times b = b \times a$
     • Example: $(-2) + (+7) = (+7) + (-2)$
     • Example: $(-3) \times (+4) = (+4) \times (-3)$

3. Associative Property

   • Addition and multiplication are associative:
     • $(a + b) + c = a + (b + c)$
     • $(a \times b) \times c = a \times (b \times c)$
     • Example: $(2 + 3) + 4 = 2 + (3 + 4)$
     • Example: $(-2 \times 5) \times 3 = -2 \times (5 \times 3)$

4. Distributive Property

   • $a \times (b + c) = (a \times b) + (a \times c)$
     • Example: $3 \times (4 + 5) = (3 \times 4) + (3 \times 5)$

5. Identity Properties

   • Additive Identity: $a + 0 = a$
   • Multiplicative Identity: $a \times 1 = a$

6. Inverse Property

   • Additive Inverse: $a + (-a) = 0$
   • Multiplicative Inverse: $a \times \frac{1}{a} = 1$ (except for zero)