Number sentences are one of the first things that primary school children are introduced to because it is one of the leading frameworks upon which most mathematics is taught.

Whether you’re solving simple addition problems or complex algebraic problems, you will utilise number sentences.

They help make sense of what calculations need to be done, and without them, you’ll just have numbers without a way to interpret them. But what exactly are number sentences?

That’s what we’ll explain in this article, along with the different types of number sentences, why understanding them is so important, and much more.

A number sentence can be defined as a mix of numbers and signs – also known as simple mathematical symbols – that presents a mathematical problem or equation which needs to be solved. As such, a number sentence is used synonymously with the phrases ‘math problem’ and ‘math equation’. Signs can be anything from operation signs such as multiplication (×) and division (÷) to an equal (=) or inequality (<>) sign.

Number sentences can include numbers and a mathematical operations sign on both sides and are often separated by an equals sign or inequality sign. 1 + 2 = 3 is considered a number sentence because it has all the necessary parts. It has numbers (1 and 2) and an addition sign (+), which is separated from another number (3) by an equal sign.

Due to the wide range of number and sign combinations to choose from, there are essentially an infinite amount of number sentences out there. However, they will typically fall into the following categories:

This is when the number sentence has an expression on one side, an equals sign, and then a number after it. To use the same example as above, 1 + 2 = 3 is an addition number sentence.

#### Subtraction number sentence

A subtraction number sentence usually follows the same format as an addition number sentence. An example of this would be 10 - 7 = 3.

#### Division number sentence

The number sentence 6 ÷ 3 = 2 is a prime example of a division number sentence.

#### Less than number sentence

This is where the format changes. Instead of an equals sign, there will be a less than (<) sign. The less than sign shows that there is an imbalance in the number sentence, where the left expression is smaller than the right expression. An example of this would be 9 + 3 < 15

We know that 9 + 3 = 12. If we substitute 12 into the number sentence, we will get 12 < 15. This holds true since 12 is indeed less than 15.

#### Greater than number sentence

The same thing applies here, except we use a greater than sign. The greater than (>) sign shows that the left expression is bigger than the right expression. An example of this would be 20 + 3 > 21.

We know that 20 + 3 = 23. If we substitute 23 into the number sentence, we will get 23 > 21. This holds true since 23 is greater than 21.

#### Fraction number sentence

Fraction number sentences can include an equals sign or inequality sign but will have fractions instead of whole numbers. For instance, ⅕ + ⅗ = ⅘.

#### Algebraic number sentence

Lastly, we have algebraic number sentences. These substitute the whole numbers with letters such as a + b = c. As with fraction number sentences, these can vary, with some having equal signs and some having inequality signs.

A valid number sentence can be both true and false and is dependent on the expressions in the number sentence. Let’s explore what each one is and how they differ from each other.

#### True number sentence

A true number sentence is one where the written sentence is correct and is balanced on both sides. This is usually shown by using an equal sign to show that one side of the equation is equal to the other side of the equation. Examples of number sentences that are true include the following:

• 1 + 1 = 2
• 10 × 5 = 50
• 22 - 7 = 15
• 27 ÷ 9 = 3
• 6 × 7 = 42

The above examples are relatively straightforward since they use the four main basic mathematical operators in an expression on one side, with the answer to the problem on the other. However, you can also have a true number sentence which consists of an expression on both sides. Calculating both expressions separately will result in equal values. For instance, suppose we have the number sentence 8 × 4 = 2 × 16.

If we solve this example problem, we will find that it is a true number sentence. 8 × 4 = 32 and 12 ×16 = 32, which we can write as 32 = 32. Since these are equal, it is a true number sentence. More examples of this are as follows:

• 9 - 8 = 1 ÷ 1
• 3 + 12 = 5 × 3
• 25 × 10 = 1,000 - 750

#### False number sentence

Whereas true number sentences are balanced on both sides of the equals sign, a false number sentence is one where it is unbalanced. This is often described as an ‘untrue’ problem. For instance, suppose we have the number sentence 5 × 4 = 15. If we calculate this problem, we will find that 5 × 4 is actually 20. 20 does not equal 15, and therefore this number sentence is defined as false.

False number sentences are typically used to test whether a student or person has a sound understanding of mathematical operations and expressions, as this deeper understanding will be required to distinguish between a false and true number sentence. More examples of false number sentences include the following:

• 4 - 2 = 3
• 64 ÷ 8 = 9
• 25 × 9 = 750 ÷ 11

Sometimes, you will find that you don’t want to write a false number sentence, but you still need to show that the number sentence is not equal. This is where using an inequality sign makes sense.

There are four inequality signs to be aware of:

• Greater than (>) – The expression or values on the open side of the sign are said to be bigger than what’s on the closed side
• Less than (<) – The same holds true for this. The expression or values on the open side of the sign is said to be bigger than what’s on the closed side
• Greater than or equal to (≥) – The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side
• Less than or equal to (≤) – The same holds true for this. The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side

These signs provide much more flexibility since they show the relationship between both sides of an expression without having to create a false number sentence – particularly, the ‘... equal to’ signs as they can make all the difference in statistics and computer science problems where you can have unknown values. Here are a few examples of inequality signs being used in number sentences:

• 9 × 9 > 55
• 55 < 9 × 9
• 23 + 4 < 29
• n + 15 ≥ 20: If you calculate this problem, it means that n must be equal to or greater than the value 5.

So far, the examples we’ve seen of number sentences have been written as numbers and symbols. But, you can also describe number sentences as math sentences, also known as written word problems.

This is where the problem is described using words instead of explicit operational signs. Many schools use math sentences to train their students on how to use their understanding of language to create a number sentence which they can then solve. An example of this would be the following sentence:

Jack has 4 apples, and Liz has 10. If Liz gives 3 apples to Jack, how many apples does she have left?

To calculate this, students would have to connect the dots and translate the words into a number sentence with numbers and operational signs. This particular problem can be solved by doing 10 - 3 = 7. The answer to this problem would be 7 apples.

In the past, teachers and educators would use the word ‘sum(s)’ when referring to number sentences. The problem with this was that sum has different meanings depending on its context.

For instance, the sum is used synonymously with the word ‘total’; you could say ‘the sum of 1 + 1 = 2.' Therefore, referring to number sentences as a sum can confuse those who are new to learning mathematics.

To correct this, in most English-speaking countries such as the USA, UK, Canada, Australia, and New Zealand, math problems are now described as number sentences – or alternatively, problems and equations. This means that when number sentences are mentioned, students immediately know what is being referenced.

Understanding how to write and use number sentences is crucial because it displays fundamental mathematical structures and forms the basis of algebraic equations, which are a key part of any curriculum.

Number sentences also allow students to learn the correct syntax, which is akin to learning the proper punctuation and grammar when writing in English. Without knowing how to write and interpret number sentences, the why behind the calculations remains unclear and can significantly hinder their math progress. This ability to record and analyze connections between numbers is an invaluable method to further your mathematics knowledge.

The benefits of learning how to use number sentences aren’t limited to the classroom. They also have real-life applications. Whether you’re in a bank looking to withdraw money or simply calculating how many ingredients you need to buy for a two-person meal, number sentences will be the framework that you use to solve the problem.