What is a number? that may sound like an unusual question, but it is interesting to investigate how a number is a value in maths that signi?es a quantity or value. Numbers form the foundation of maths; we use them for counting and measuring.

 

What Is a Number?

We use numbers every single day of our lives, so it is often very easy to overlook the question of what a number is. So, a number is a value used in maths to represent a quantity or a value. We use numbers for counting, for measuring, and for labelling. Basically, without numbers, we don't have maths! We see number when we are telling the time, buying something from a shop, or even the score while watching a sporting match. We use numbers for our date of birth, to phone each other, and to measure our height. Numbers are literally a fundamental part of everything around us.

 

Where do Numbers Come From?

We don’t need to go into too much detail about the history of numbers, but it is interesting to learn where numbers come from. There is evidence to show that early humans used different symbols to represent values. This suggests that humans have been using numbers for a long time.

 

There are even different ways to reference numbers. For example: a tally chart involves drawing up to four vertical lines next to each other, with the ?fth line being drawn diagonally across them. However, tally charts are not very easy to use when dealing with large quantities of numbers! It wasn't until roughly during the 7th century that a decimal positional method came into existence, stemming from India. This decimal position method, or base ten method, consisted of ten different symbols used to represent a value or quantity. This method spread from India to Europe and is known as the Hindu-Arabic numeral system. The symbols used are the symbols that we now use today as numbers, which of course are: 0, 1, 2, 3, 4, 5 6, 7, 8, and 9.

 

The Decimal Number System

The number system that we use today consists of the numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This number system is known as the Decimal Number System. It is a way of writing down numbers in a logical way. We are able to use these ten numbers to form other numbers with a higher value. In fact, there are literally in?nite numbers that we can create by using these numbers in any combination.

 

Numbers in Alphabetic Form

As well as using symbols to represent numbers, we can also simply write the numbers out in written form using letters. For example, instead of writing the number 1, we could write the word "one", instead of writing the number 2, we could write the word "two". Each value has a written representation as well as a numerical one.

 

How to write numbers in words in English:

When writing the numbers in English we always write the larger part of the number ?rst and work our way through the number to the smaller digits. This matches our general reading style of working left to right and is easier to follow than presenting numbers in a random and unstructured order. Two thousand, three hundred and four is much easier to understand than four and two thousand and three hundred, something that might confuse people occasionally, is that when writing numbers in words, we use the singular form for hundreds, thousands, millions etc. For example, we would write 'three hundred' instead of 'three hundreds'.

 

What is a digit?

A digit is any of the numerals from 0 to 9 and is used to represent, whole, decimal, positive and negative numbers. A digit is a numerical representation of the numbers 0–9. By placing these digits in different positions within a number you can show any whole, decimal, positive, or negative number imaginable. Digits are used together to represent a variety of  numbers and values. When you come across references to 3, 4, or 5-digit numbers, that is referring to how many digits are needed to make that number. A 3-digit number will be made up of 3 digits, a 4-digit number of 4 digits, a 5-digit number of 5 digits and so on.

 

Is a digit the same as a number?

Digit, number, numeral, numerical representation are all terms that you may come across, but there are key differences between them.

Numbers can be expressed in different ways, through ?ngers, clicks, words, sounds, etc. A

number represents a value or amount. The ways in which a number can be shown links to its numerical representation – or how it is being expressed.

A numeral is a chosen method of numerical representation and does not always involve digits, for example with Roman numerals.

Digits are used to represent numbers.

Without an understanding of place value, we would not be able to interpret the true meaning of digits. Place value is where the position of the digit changes the value of the digit and in turn the overall number.

 

For example: The digit 7 in 937 represents 7 ones as it is in the ones place. If you were to change the position of the 7 within that number, you would change the number. If you swap the 7 with the digit in the middle, the number will become 970 and the 7 would now represent 7 tens, giving it a value of 70. If we were to move the 7 again and swap it with the digit 9 our new number would be 790. Our delightful digit has changed in position and value once again. The move to the hundreds place means that our small 7 is now worth 700.

 

Meanings of All Math Symbols

Below if a table containing the meanings of all of the most commonly used math symbols here, we take  a look at the mathematics symbols, their names and meanings. We will then at some in  more depth.

 

Symbol	Name	Meaning
=	Equals sign	equality
+	Plus sign	addition
−	Minus sign	subtraction
×	Times sign	multiplication
*	asterisk	multiplication
÷	division sign / obelus	division
/	division slash	division
.	Period / decimal point	Decimal separator
≠	not equal sign	Inequality
≈	approximately equal	approximation
>	Strict inequality	Greater than
<	Strict inequality	Less than
≥	Inequality	Greater than or equal to
≤	Inequality	Less than or equal to
[ ]	brackets	calculate expression inside ?rst
√a	Square root	√a ⋅ √a = a
x	x variable	unknown value to ?nd
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle
°	degree	1 turn = 360°

What are mathematics symbols?

There are lots of different mathematics symbols out there and they all have different meanings. Symbols are used to save space and time when writing, in the same way that number symbols save time and space compared to writing out the written form of a number. Mathematical symbols are small icons that represent actions. For example, it is much quicker to write 14 × 5 than "fourteen multiplied by ?ve".

 

Symbol meaning: The four operations.

The most obvious place to start with mathematical symbols is with the four operations. These are addition, subtraction, multiplication, and division. Each operation (also known as a process) has its own symbol that can be used in number sentences:

Addition uses the plus symbol +

Subtraction uses the minus symbol  –

Division is shown with the divide symbol  ÷

Multiplication is represented by the multiplication symbol ×

As shown in the table sometimes these symbols can be alternated to use an easily accessible symbol when typing on a computer.

 

Symbol meaning: Greater than and lesser than

There are also mathematical symbols which show the relative size of a group of numbers. These are the greater than and less than symbols: < and >. Greater than and less than symbols show the  relationship between two numbers. They are used to show which numbers have a higher or lower value. They are sometimes referred to as 'crocodile symbols', since they look like the open mouth of a crocodile. The open mouth points towards the greater number in the pair or sequence, because hungry crocodiles will always eat the bigger meal!

For example:

72 > 37

123 < 150

 

 

Symbol meaning: squares and square roots

A commonly seen symbol that you come across is the squared symbol. This is a small number 2 written in line with the top of a number.

For example:

2²

6²

10²

The square symbol simply means that a number needs to be multiplied by itself.

 

You could write it out in longform as, for example:

2 × 2 = 4

6 × 6 = 36

10 × 10 = 100

However, the squared symbol is much quicker and easier to write. You can see it in lots of maths equations to show that a number should be multiplied by itself. The opposite of the squared symbol is the square root symbol. This looks like a tick and is written over a number, like this: √4

 

 

What is a Whole Number?

The de?nition of a whole number is simply any positive number that does not include a fractional or decimal part. This means that, for example, the numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all whole numbers. Numbers such as -3, 2.7, or 3 ½ are not whole numbers.

Whole numbers can also be known as integers. There are 3 main factors to consider when working out whether a number is a whole number.

1.  A whole number must be positive, not a negative number, (also known as a minus number). This means that it has to be of a value of 0 or higher. For example,

0, 1, 2, and 3 are all whole numbers, however, -1, -2, and -3 are not.

2.  A whole number cannot include any fractional element. That means that numbers such as 1 ½, 3 ¼, and 7 ? are not whole numbers, but 1, 3, and 7 are.

3.  A whole number cannot include any decimal element. This means that numbers such as 3.4, 7.9, and 11.234 are not whole numbers, but 3, 7, and 11 are.

 

What is an integer?

Integer numbers are whole numbers that can be positive or negative, but do not have fractions or decimals. You can use the term "integer" to describe a whole number. Unlike whole numbers, integer numbers refer to negative numbers as well as positive amounts. Is 0 an integer number? Different people de?ne integers in different ways. Some people would argue that negative numbers are integer numbers, whereas others would claim that even number 0 is not an integer.

Usually, though, 0 is an integer, as are 109, 7, -3, 8 and -28, because these are whole numbers with no fractions or decimals.

 

Rounding to the Nearest Whole Number

Rounding to the nearest whole number means ?nding the whole number that is most similar in value to a given non-whole number. When rounding to whole numbers, we are aiming to round the number to the nearest 10, 100 or 1000 and so on. Learning this can make calculations a lot easier, so it’s useful knowledge to have!

Here are some examples:

The nearest whole number to 3.2 is 3.

The nearest whole number to 5.7 is 4.

The nearest whole number to 2 ¼ is 2.

The nearest whole number to 8 ? is 9.

This can get tricky with numbers such as 2.5 or 4 ½ because they are exactly between two whole numbers. We have an entire subheading for this topic that will go into rounding in a lot more detail.

 

What are counting numbers?

Counting numbers are whole numbers that are used to count things. These could be 1, 2, 3, 4, etc. They're a type of number that you can use to say how much you have counted of something. They are any positive number. However, you cannot have counting numbers that are negative. This is because you can't have a negative number of something. This also doesn't include fractions or decimals. Counting numbers can only ever be whole positive numbers. When it comes to counting numbers, there's been some debate around whether the number zero is classed as a counting number. Whilst you can't have a negative amount of something, you can have none of it. Which would be a zero amount of whatever you're counting. For this reason, some people think that zero should be classed as a counting number. There is no speci?c rule declaring that zero isn't a counting number, but it depends on each person individually.

 

Different Ways to Write Whole Numbers

There are different ways of writing whole numbers, and they're each a de?nition of whole number forms. These are known as 'number forms'. The six different ways we can write a whole number are listed, below:

Standard form

Word form

Short word form

Expanded form

Exponential form

Scienti?c form

Roman numerals should also be put in their historical context, so pupils understand that there have been different ways to write whole numbers and that the important concepts of 0 and place value were introduced over a period of time.

 

Visual Aids to Explain Whole Numbers

When explaining the difference between whole numbers and non-whole numbers, visual aids such as counters and shapes can be useful.

For example, you can show a whole counter to represent a whole number, and part of a counter to show non-whole numbers. You could also do this with money. Try using whole pounds to represent whole numbers, and pence to represent decimal values between whole numbers.

For example, £3 would represent a whole number, whereas £3.20, for instance, would represent a non- whole number. A number line can also be a helpful tool.

What is the Mental Method?

Mental maths is the process of working out maths calculations and carrying out problem-solving mentally, without the need to write down any working out. Children will learn need to complete mental subtraction, addition, and other mental calculations, which involve using speci?c techniques for solving different types of problems rather than memorizing the answers to equations, using a calculator, or using written methods for workings.

 

Why is mental maths important?

Why is mental maths important? basically, it helps to develop problem- solving skills that promote faster calculations. It also allows us to see the relationship between numbers and the patterns they make. As learners progress, they'll also be expected to complete more complex calculations in their head, so getting a grip of mental maths techniques at an early age will help encourage this process. Without the ability to do mental maths, it can be di?cult to complete ordinary daily tasks such as counting money, taking measurements, and even calculating time.  Mental maths is the key to con?dence and ?uency.

 

What are the core parts of mental maths skills?

There are some important skills that children will gain from practicing mental maths:

Recalling facts - Mental maths is a great test of memory and regular practice will ensure that some mathematical problems are second nature to learners. This is a foundational element of mathematics, which will stand them in good position for more complicated sums later on.

Mathematical speed - It's all about speed! There are lots of reasons why speed is good for doing sums.

Estimating calculations - Learning how to estimate the answer is a brilliant, real- world skill that children can learn as part of mental maths. They can do this by rounding numbers up or down before they do any sums.

 

Mental Maths Methods

There are lots of different strategies we can use to help us do mental maths calculations. Here are a few examples:

 

Mental Subtraction

Counting Back - This is a useful strategy to apply when subtracting smaller numbers. Children will be taught to use it together with a number line or a number square at ?rst, but then they will be able to complete it mentally. Allowing children to visualise numbers on a number line is one step closer to them being able to work out the entire mental maths problem in their heads.

 

For example: If we want to subtract 12 from 87, we can take the smaller