Learn the 5 properties of addition with examples: commutative, associative, identity, inverse and distributive. For elementary and middle school with exercises.
The 5 properties of addition: 1) Commutative: a+b=b+a. 2) Associative: (a+b)+c=a+(b+c). 3) Identity: a+0=a. 4) Inverse: a+(-a)=0. 5) Distributive: a×(b+c)=ab+ac.
Order doesn't matter. 3+5 = 5+3 = 8. Useful for mental math.
Grouping doesn't matter. (2+8)+5 = 2+(8+5) = 15. Group numbers that make 10.
Adding zero doesn't change the number. 7+0=7. Zero is the additive identity.
Every number plus its opposite equals zero. 5+(-5)=0.
3×(4+5) = 3×4+3×5 = 27. Connects addition and multiplication.
These properties also apply in algebra. Visit propiedades de la suma for the Spanish version.
Adding $12 + $8 or $8 + $12 gives the same total of $20. Order of purchase doesn't change the total.
47+38+62: regroup as (47+53)+38=100+38=138 instead of 47+38+62 in order.
Depositing $0 to a $500 account: $500+$0=$500. Balance unchanged.
Yes, all properties of addition apply to any real number including decimals, fractions and negative numbers.
No. 10−3=7 but 3−10=−7. Subtraction is NOT commutative. Only addition and multiplication have the commutative property.
Commutative: reorder → 3x+2x+5+1. Associative: (3x+2x)+(5+1)=5x+6.
Distributive: 4x+4×3=4x+12. This is the foundation of algebra.
No. 8−3=5 but 3−8=−5. Only addition and multiplication are commutative.
Because adding it to any number leaves the number unchanged — it does not alter the identity of the number. 7+0=7, x+0=x always.
Yes. All properties of addition apply to any real number: integers, decimals, fractions, negatives and even irrational numbers like π.
a+b = b+a. Order does not change the sum. 5+3 = 3+5 = 8.
(a+b)+c = a+(b+c). Grouping does not matter.
a+0 = a. Adding zero does not change the number.
a+(−a) = 0. Every number has an opposite that cancels it.
These properties are not just abstract rules — they explain why calculations work. The commutative property lets you reorder to add easier numbers first. The associative property lets you group numbers strategically.
The commutative and associative properties together let you rearrange and regroup numbers to make calculation easier. When adding a list of numbers, always look for pairs that make nice round numbers: 38+62=100, 47+53=100.
La suma de 1 a 100 = 5,050 fue calculada por Gauss de niño emparejando: 1+100=101, 2+99=101... hay 50 pares. 50×101=5,050. La propiedad asociativa hace este tipo de reagrupamiento posible.
La propiedad clausura garantiza que la suma de dos números del mismo conjunto da otro número del mismo conjunto. Naturales+Naturales=Natural. Enteros+Enteros=Entero. Reales+Reales=Real. Pero √2+√2=2√2 (irracional+irracional puede ser racional).
Commutative: a+b=b+a 3+5=5+3=8
Associative: (a+b)+c=a+(b+c) (2+3)+4=2+(3+4)=9
Identity: a+0=a 7+0=7
Inverse: a+(−a)=0 5+(−5)=0
The commutative and associative properties let you reorder and regroup numbers to simplify mental math. Always look for pairs that sum to 10, 100, or 1000 first — this is the key to fast addition.