Key takeaways
- Multiplication is repeated addition: Product = a × b.
- In long multiplication, multiply by the ones digit, then the tens digit (shifted one place left).
- Add the partial answers together to get the final product.
- Multiplication is commutative: a × b always equals b × a.
In this guide
What is multiplication?
Multiplication combines equal groups into a total. If you buy 5 bags with 7 apples in each, multiplying 5 × 7 tells you the total number of apples without counting one by one.
Why does multiplication matter?
Multiplication turns slow, repeated addition into one fast step. Instead of adding 7 five times, you multiply once. It's also the foundation for area, scaling recipes, and compound growth like interest on savings.
5 × 3 means five groups of three, giving 15 — not 5 + 3, which is only 8. Mixing these two up is the single most common early math mistake.
The formula
| Symbol | What it means | Example |
|---|---|---|
| Product | The answer after multiplying | 322 |
| a | The first number (a factor) | 23 |
| b | The second number (a factor) | 14 |
How to multiply step by step (long multiplication)
- 1Stack the numbers. Write the bigger number on top and the smaller number underneath, lined up by place value: 23 over 14.
- 2Multiply by the ones digit. Multiply the top number by the bottom number's ones digit:
23 × 4 = 92. - 3Multiply by the tens digit, shifted left. Multiply the top number by the bottom number's tens digit, then shift the result one place left (add a zero):
23 × 10 = 230. - 4Add the partial answers.
92 + 230 = 322. That's your final product.
Try 23 × 14 (or your own numbers) in the calculator below and watch each partial product appear.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Losing track of carried digits. If a single-digit product is 10 or more, carry the extra digit into the next column, just like in addition.
- Confusing multiplication with addition. 5 × 3 means five groups of three (15), not 5 + 3 (8).
- Forgetting a placeholder zero. The second row of long multiplication always starts one column to the left, which the shifted zero represents.
Tips and tricks
- Break big numbers apart. 23 × 14 is the same as
(20 × 14) + (3 × 14) = 280 + 42 = 322, which some people find easier mentally. - Check your work by dividing the product by one factor — you should get the other factor back.
- Multiplying a number by itself repeatedly leads to exponents and powers, and multiplying two side lengths gives you the area of a rectangle.
Where you'll use it in real life
- Shopping: working out the total cost of several identical items.
- Cooking: scaling a recipe up for more people by multiplying every ingredient.
- Home projects: finding the area of a room by multiplying its length and width.
- School: building times-table fluency, which makes every later math topic faster.
Quick summary
To multiply two numbers by hand, use long multiplication: multiply by the ones digit, then by the tens digit (shifting one place left), and add the partial results together. Multiplication is commutative, so the order of the factors never changes the answer. Try the calculator above to see every partial product for your own numbers.
Worked example
A school orders 23 boxes of pencils, with 14 pencils in each box. How many pencils did the school order in total?
- 01Product = a × b
- 02Product = 23 × 14
- 03Product = 322
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the easiest way to multiply two-digit numbers by hand?
Use long multiplication: multiply the top number by each digit of the bottom number separately, shifting each row one place left as you move to the next digit, then add all the rows together.
Does the order of multiplication matter?
No. Multiplication is commutative, so a × b always gives the same answer as b × a. 4 × 6 and 6 × 4 both equal 24.
How do I check a multiplication answer is correct?
Divide the product by one of the original numbers — you should get the other number back. For example, 322 ÷ 23 should give 14.
What's a quick way to multiply by 10?
Just add a zero to the end of the number. 23 × 10 = 230, because multiplying by 10 shifts every digit one place to the left.
