# Multiplication chart

A multiplication chart, also known as a multiplication table, or a times table, is a table that can be used as a reference for the 100 multiplication facts. It can be helpful for learning and memorizing multiplication facts, which is essential, since multiplication is used throughout all areas of mathematics in some form or another. Being familiar with all the multiplication facts enables a person to focus on more complex mathematical concepts that involve multiplication, without having to worry about the actual multiplication.

× | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |

4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |

5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |

6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |

7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |

8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |

9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |

10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

### How to use a multiplication chart

To use a multiplication chart, first look at the rows and columns in grey in the figure above. Rows are read horizontally from left to right while columns are read vertically from top to bottom. The green diagonal on the chart represents the squares of the numbers, namely 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc.

Use the green diagonal to understand how the multiplication chart is read. The value in the green diagonal is the product of the column and row values that align with it in grey. For example, 25 is the product of 5 and 5. 20 is the product of 4 and 5 and 5 and 4, etc.

In other words, to use the multiplication chart, choose the two values that you want to multiply from the grey row and column, then determine what value an imaginary horizontal and vertical line (drawn from the grey column and row respectively) would intersect at to determine the product of the two values.

## Memorizing the multiplication chart

Although 100 facts may seem like a large number to memorize when just starting to learn multiplication, the number of facts that need to be memorized can be reduced by using certain properties of multiplication.

### Commutative property of multiplication

The commutative property of multiplication states that the order of multiplication doesn't matter. Given two numbers, a and b:

a × b = b × a

We can confirm this by looking at the multiplication chart and seeing that regardless of whether we look at the multiplication fact 2 × 8 = 16 or 8 × 2 = 16, the solution is still 16. This is true for anything being multiplied. Since the order doesn't matter, we only really need to memorize numbers below or above (and including) the diagonal shown in green on the chart. This property almost halves the number of multiplication facts we need to memorize.

### Identity property of multiplication

The identity property of multiplication states that any number a multiplied by 1, is equal to a:

1 × a = a

Since 1 multiplied by any number is that number, as long as we know this property, it is not necessary to memorize the first row or column of the multiplication chart.

### Multiplication by 10

Due to the nature of the decimal numeral system, multiplying any integer by 10 results in that same integer with a 0 added to the end. For example, 2 × 10 = 20, 20 × 10 = 200, 200 × 10 = 2000, and so on. It doesn't matter what the number is, multiplication by 10 results in a shift of the decimal place one position to the right. Since integers have no decimal values, this just means adding a 0 (For decimals, the decimal point is moved to the right one place). This allows us to remove the last row and column of the multiplication chart, leaving the following 36 multiplication facts that need to be memorized.

× | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | ||||||||||

2 | 4 | |||||||||

3 | 6 | 9 | ||||||||

4 | 8 | 12 | 16 | |||||||

5 | 10 | 15 | 20 | 25 | ||||||

6 | 12 | 18 | 24 | 30 | 36 | |||||

7 | 14 | 21 | 28 | 35 | 42 | 49 | ||||

8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | |||

9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | ||

10 |