Knowledge base · Maths - grade 4Maths - grade 4
Natural numbers. The number line
Digits and place value, comparing numbers, natural numbers and the number line - with step-by-step examples for year 4.
You use numbers from morning to night: on a clock, in a shop, on a bus, in a match score. And all of them, even the really big ones, are written with just ten symbols. In this lesson you will see how that is possible, how to tell at a glance which of two numbers is bigger, and how to line up all the numbers in order on a single straight line - the number line.
What is the difference between a digit and a number?
Digits are the symbols we write numbers with. There are exactly ten of them:
0 1 2 3 4 5 6 7 8 9
Digits are like letters, and numbers are like words. You build words out of letters, and numbers out of digits. The number 7 is written with one digit, 305 with three, and 2 026 with four.
A digit is a single symbol. You write a number with one digit or with several.
Why exactly ten digits? It is thought to be because people have counted on their fingers for a very long time, and we have ten fingers. That is where the name of our way of writing numbers comes from: the decimal system, from the Latin decem, meaning ten.
How many digits does the number 2 026 have? And how many different digits were used to write it?
Hint: Digits can repeat. Count them twice: once all of them, once only the different ones.
Show the answerHide the answer
Answer
The number 2 026 has four digits: 2, 0, 2 and 6. Only three of them are different: 2, 0 and 6, because the 2 appears twice.
Why can the same digit stand for 5, 50 or 500?
Look at the number 555. The same digit three times, yet each 5 means something different. The first from the right is five ones. The second is five tens, which is 50. The third is five hundreds, which is 500.
A 5 is always a five, but the number it stands for depends on the place where it stands. That is why our way of writing numbers is called a place value system.
Each place is worth ten times as much as the place to its right: ten ones make one ten, ten tens make one hundred, ten hundreds make one thousand. See what each digit of 3 725 stands for:
| thousands | hundreds | tens | ones |
|---|---|---|---|
| 3 | 7 | 2 | 5 |
| 3 000 | 700 | 20 | 5 |
What is zero for?
Zero looks as if it means nothing, but it has a very important job: it holds a place. In 305 the zero says there are no tens, and thanks to it the 3 stays in the hundreds place. Cross it out and you are left with 35 - the 3 drops into the tens place and stands for only 30.
Big numbers are split into groups of three
The longer the number, the more places it has. So as not to get lost among them, we write big numbers with a space after every three digits, counting from the right. The first group of three from the right is the ones, tens and hundreds, the next one is the thousands, and the one after that is the millions.
| hundred thousands | ten thousands | thousands | hundreds | tens | ones |
|---|---|---|---|---|---|
| 2 | 5 | 0 | 3 | 1 | 8 |
| 200 000 | 50 000 | 0 | 300 | 10 | 8 |
There are three 4s in the number 4 404. What number does each of them stand for?
Hint: First work out which place each 4 stands in. The zero in the tens place counts too.
Show the answerHide the answer
Answer
The first 4 from the left is in the thousands place, so it stands for 4 000. The second is in the hundreds place - it stands for 400. The last one stands for 4 ones. The zero says there are no tens at all.
How do you compare two numbers?
Three signs are used for comparing numbers:
We read the first as “seven is greater than three”, the second as “three is less than seven”, and the third as “five equals five”.
How do you not mix up > and <? The sign always opens towards the bigger number and points its sharp end at the smaller one.
First, count the digits
If one number has more digits than the other, it is the bigger one - and you do not even need to look at which digits they are. Why? Because the smallest four-digit number, 1 000, is still bigger than the biggest three-digit one, 999.
Then compare the digits from the left
When the numbers have the same number of digits, you compare them from the left, place by place. The first digit that differs decides everything:
Why is that one digit enough, while the ones after it do not count? Because one extra hundred outweighs anything that can stand in the tens and ones places. At most 99 fits there, and that is still less than 100.
The most common mistake
Some people look at which number has “bigger digits”, for example lots of nines. That is how this mistake happens:
incorrect
correct
The thousands are equal - 5 each. But in the hundreds place 5 099 has a 0, while 5 100 has a 1. That settles it, and the nines stand further to the right, where they have no say any more.
What are natural numbers?
The numbers you count objects with - zero, one, two, three and so on - are called natural numbers:
0, 1, 2, 3, 4, 5, 6, 7, …
The smallest natural number is 0. There is no biggest one: you can add 1 to any natural number and get an even bigger one.
In Polish schools zero belongs to the natural numbers. That is not the case everywhere - in many textbooks around the world the natural numbers start from 1. It is a convention, not a discovery.
What is a number line?
All the natural numbers can be lined up in order on a single straight line. Such a line is called a number line. To draw one you need three things:
- the point 0 - that is where you start;
- a unit segment, that is the segment from 0 to 1 - you choose its length yourself, for example 2 squares in your exercise book;
- an arrow at the end - it shows which way the numbers grow.
To get to the number 4, you mark off four unit segments from zero. To get to 9 - nine. That is why all the gaps between neighbouring ticks must be equal: if one of them were longer, the numbers after it would end up in the wrong places.
You can read straight off the line which number is bigger: the bigger one always lies to the right of the smaller one.
How do you read the distance between numbers on the line?
The distance between two numbers on the line is the number of unit segments between them. Between 3 and 8 there are five:
You do not have to count them one by one. Just subtract the smaller number from the bigger one:
What is the distance on the number line between 12 and 30?
Hint: Subtract the smaller number from the bigger one.
Show the answerHide the answer
Answer
There are 18 unit segments between 12 and 30, so the distance is 18.
What if the numbers are big?
A line with the numbers up to 10 fits in any exercise book. But what about 80? A square in an exercise book is half a centimetre wide. If the unit segment were 1 square long, you would need 80 squares to reach 80, which is 40 centimetres - more than the width of the book.
That is why we do not always mark every number on the line. You can put a tick, for example, at every 5 - then the segment between neighbouring ticks stands for five units, even though it is only 2 squares long.
In the same way, the segment between ticks can stand for 10, 20 or 100 units. Always check first what one gap is worth, and only then count the ticks.
Click the tick on the line where this number lies: 7
Worked examples, step by step
Which numbers do the marked points stand for?
The points A, B and C are marked on the number line. Which numbers do they stand for?
- 1
Find two labelled numbers and count the gaps between them. From 0 to 100 there are 5 gaps.
- 2
Divide to find out what one gap is worth:
- 3
Point A lies 3 gaps after zero:
- 4
Point B lies 2 gaps after 100. It pays to count from the nearest labelled tick, not from zero:
- 5
Point C is also easiest to count from 100 - it lies 4 gaps further on:
Compare the numbers
Put the right sign between the numbers: >, < or =.
- 1 010 and 999
- 45 781 and 45 718
- “three hundred and five” and 350
- 6 070 and 6 070
- 1
1 010 has four digits, and 999 has three. More digits means a bigger number:
- 2
45 781 and 45 718 both have five digits, so you compare them from the left:
457814571845 781 > 45 718The digits from the left match until the tens place: 8 > 1. - 3
First write “three hundred and five” in digits: 3 hundreds, 0 tens and 5 ones, that is 305. Now compare it with 350:
305350305 < 350The digits from the left match until the tens place: 0 < 5. - 4
6 070 and 6 070 have the same digit in every place, so they are equal:
Draw a number line and mark the numbers
Draw a number line, choose a suitable unit and mark the numbers 2, 5 and 9 on it. Then draw a second line - for the numbers 20, 50 and 80.
- 1
The biggest of 2, 5 and 9 is 9. Let the unit segment be 2 squares long. Then you need 18 squares to reach 9, which is 9 centimetres - it will fit.
- 2
Mark 0 and 1, keep marking off unit segments and put points at 2, 5 and 9:
- 3
On the second line the biggest number is 80. With a unit segment 2 squares long you would need 160 squares, which is 80 centimetres - far too much. So let 2 squares stand for 10 units.
- 4
You put a tick at every 10. There are 8 gaps up to 80, which is 16 squares - 8 centimetres:
Where are natural numbers and the number line useful?
- A ruler and a tape measure are pieces of a number line: zero at the start, equal spacing and numbers growing in one direction. When you measure something with a ruler, you read a number off a number line.
- A timeline in a history lesson is a number line with years instead of ordinary numbers. The distance between two dates is the number of years that passed between them.
- In a shop you compare prices: a bike for 1 299 zł is cheaper than one for 1 350 zł, because 1 299 < 1 350.
- Money shows place value: 375 zł is 3 notes of 100 zł, 7 notes of 10 zł and 5 coins of 1 zł.
- In games and competitions you compare scores: 12 450 points is more than 12 405, even though both numbers start the same way.
- Not every digit stands for an amount. In a phone number or a PIN, digits are just symbols, and there is no point asking which one stands for more. That shows nicely the difference between a digit and a number.
A puzzle to finish
You have four cards with the digits 3, 0, 7 and 5. You make four-digit numbers out of them, using each card exactly once.
What is the biggest and what is the smallest number you can make?
Show the solutionHide the solution
Solution
The biggest is 7 530: in the thousands place, which is worth the most, you put the biggest digit, followed by smaller and smaller ones. The smallest is 3 057, not 0 357! A number cannot start with a zero - 0357 is just 357, a three-digit number. So the smallest digit other than zero, 3, goes first, and the zero comes right after it.
Remember
There are ten digits and infinitely many natural numbers. A digit’s value depends on the place where it stands. You compare numbers first by how many digits they have, then digit by digit from the left. On the number line the bigger number lies to the right of the smaller one, and the distance between two numbers is their difference: the bigger minus the smaller.
PhiBoard is a free online whiteboard for maths
This lesson is part of it. On the board itself you can draw a number line, a graph or a figure, write a formula and solve a problem together with a teacher or a friend - live, in the browser. Our app is completely free.
Check whether you really know it
Theory is half the way. Work through a few exercises on the same material - each with a step-by-step solution, no account and no fees.
Frequently asked questions
Is zero a natural number?
In Polish schools, yes - zero counts as a natural number, so the natural numbers are 0, 1, 2, 3 and so on. It is a convention, not a discovery: in many textbooks around the world the natural numbers start from 1, and the set that includes zero is called the whole numbers.
What is the difference between a digit and a number?
A digit is one of ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. A number is the value we write with those symbols - with one digit, like 7, or with several, like 305. Digits are like letters, and numbers are like words.
What is a number line?
It is a straight line on which numbers stand in order at equal spacing. To draw one, you mark the point 0, a unit segment (from 0 to 1) and an arrow showing which way the numbers grow. A bigger number always lies to the right of a smaller one.
How do you compare two numbers with several digits?
First count the digits: the number with more of them is bigger. If they have the same number of digits, compare them from the left - the first digit that differs decides. For example, 4 352 < 4 532, because the thousands are equal and in the hundreds place there is a 3 and a 5.
How do you find the distance between numbers on a number line?
Subtract the smaller number from the bigger one. The distance between 3 and 8 is 8 − 3 = 5, that is, five unit segments.