How Do Addition and Subtraction Work?
The two most fundamental math operations — combining things together and taking them apart.
Putting Together and Taking Apart
Addition is combining two or more numbers to find their total. If you have 3 apples and get 4 more, you now have 3 + 4 = 7 apples. Subtraction is the opposite — finding what's left when you take some away. If you have 7 apples and eat 2, you have 7 − 2 = 5 left. These two operations are the foundation of all mathematics. Every other operation — multiplication, division, algebra, calculus — builds on them.
Number Lines and Counting
A number line is one of the best tools for understanding both operations. To add, start at the first number and jump right. To subtract, start at the first number and jump left. 5 + 3: start at 5, jump 3 to the right, land on 8. 9 − 4: start at 9, jump 4 to the left, land on 5. Number lines make abstract math visual and concrete.
Regrouping (Carrying and Borrowing)
When adding larger numbers, sometimes a column adds up to 10 or more. In 47 + 35, the ones column gives you 7 + 5 = 12. You write the 2 and carry the 1 to the tens column: 4 + 3 + 1 = 8. Answer: 82. This is called regrouping. Subtraction uses the reverse: in 52 − 28, you can't take 8 from 2, so you borrow 1 ten from the 5 (making it 4) and add 10 to the ones column (making it 12). Now 12 − 8 = 4, and 4 − 2 = 2. Answer: 24.
Mental Math Strategies
Make a ten: To add 8 + 7, think "8 + 2 = 10, and 7 − 2 = 5, so 10 + 5 = 15." Count up for subtraction: For 15 − 9, think "9 + ? = 15" — the answer is 6. Break apart numbers: For 36 + 47, try (36 + 40) + 7 = 76 + 7 = 83. These strategies are faster than counting on fingers and build number sense that helps with every future math topic.
The Relationship Between Addition and Subtraction
Addition and subtraction are inverse operations — they undo each other. If 5 + 3 = 8, then 8 − 3 = 5 and 8 − 5 = 3. This relationship is called a fact family, and understanding it is a breakthrough moment for young learners. It means every addition fact automatically gives you two subtraction facts — and checking subtraction with addition (or vice versa) is one of the best ways to verify your work.
The plus sign (+) that we use for addition was first used in print in 1489 by a German mathematician named Johannes Widmann. Before that, people wrote the Latin word "et" (meaning "and") or used the letter "p" (for "plus"). The minus sign (−) appeared around the same time. These symbols are now so universal that they're understood in every country on Earth, regardless of language — making math one of the only truly global communication systems.
Last reviewed: April 2026