Numbers - Session 2

Let us say there are two light-bulbs. One blinks every four seconds and other blinks every six seconds. Can you tell how many times will they blink together at the same time in one minute? Well, to answer it first you need to understand the concept of common multiples. Let us begin by understanding what a multiple is. A multiple is what you get when you multiply a number by one, two, three, and so on. For example, the multiples of three are three, six, nine, twelve, fifteen, and it keeps going. If we take another number like four, its multiples are four, eight, twelve, sixteen, twenty, and so on. These are simply like the multiplication tables you learn in school.

Now let us talk about common multiples. These are numbers that appear in both lists of multiples. Let us look at three and four again. The multiples of three are three, six, nine, twelve, fifteen, eighteen, twenty-one. The multiples of four are four, eight, twelve, sixteen, twenty. You can see that twelve is in both lists, so twelve is a common multiple of three and four.

Let us try another pair of numbers. If we take five and six, the multiples of five are five, ten, fifteen, twenty, twenty-five, thirty, thirty-five. The multiples of six are six, twelve, eighteen, twenty-four, thirty, thirty-six. You can see that thirty appears in both lists. So, thirty is a common multiple of five and six.

Some numbers have many common multiples. For example, take two and four. The multiples of two are two, four, six, eight, ten, twelve, fourteen, sixteen and so on. The multiples of four are four, eight, twelve, sixteen, twenty, and so on. So, four, eight, twelve, sixteen, twenty, and more are common multiples. They show-up in both lists, so we call them common.

You do not need to write a long list to find a common multiple. For example, try six and eight. The multiples of six are six, twelve, eighteen, twenty-four. The multiples of eight are eight, sixteen, twenty-four. You can stop at twenty-four because it shows up in both lists. So, twenty-four is a common multiple of six and eight.

Let us talk about the LCM, which stands for Least Common Multiple. This means the smallest number that two numbers have in common when you list their multiples. For example, if we take four and five, the multiples of four are four, eight, twelve, sixteen, twenty, and so on. The multiples of five are five, ten, fifteen, twenty, twenty-five. The first number that appears in both lists is twenty, so the Least Common Multiplier of four and five is twenty.

Here is another one. Try four and six. The multiples of four are four, eight, twelve, sixteen, twenty. The multiples of six are six, twelve, eighteen, twenty-four. What number comes first in both lists? That is right, it is twelve. So, the LCM of four and six is twelve. The LCM is helpful when you want things to match-up or synchronize. For example, if one drum beats every four seconds and another beats every six seconds, they will beat together after twelve seconds. That twelve seconds is their least common multiple.

Common multiples are not simply for school. They are helpful in real life too. Imagine two lights blinking. One blinks every four seconds, and the other blinks every six seconds. To know when they will blink together, you find the common multiple of four and six. Their least common multiple is twelve, so they blink at the same time every twelve seconds. This shows how simple math can solve real problems. Can you tell how many times these lights will blink together in one minute?.

Let us now understand rational numbers. A rational number is any number that you can write as a fraction. That means it can be written as one number divided by another, like three over four or five over one. Whole numbers like two or seven are also rational because you can write them as two over one or seven over one. Rational numbers can also be decimals that stop or repeat, like zero point five or zero point three three three.

Here are some examples of rational numbers. One-half is rational because it is a fraction. Four is rational because it is the same as four over one. Zero point seven five is rational because it is the same as three over four. Repeating decimals like zero point six six six are also rational because they go-on in a pattern.

You might be surprised to know that whole numbers are also rational numbers. That is because you can write the number four as four divided by one. You can write the number zero as zero divided by one. Big numbers also, like one hundred, can be written as a fraction. As long as the bottom number is not zero, the number is rational.

Now let us talk about decimals. Some decimals are also rational. If a decimal stops, like zero point five or zero point two five, then it is rational. These are called terminating decimals because they end. You can also write these as fractions, so they are also rational.

Some decimals do not stop but they keep repeating the same pattern. These are also rational numbers. For example, zero point three three three, with threes going on forever, is rational. When the repeating part is longer also, like zero point one four two eight one four two eight, it is also rational as long as it follows a pattern. All repeating decimals can be changed into fractions.

Another way to check if a number is rational is to ask-yourself this, can I write it as a fraction where both the top and bottom are whole numbers? If yes, then it is a rational number. The bottom number cannot be zero because dividing by zero is not allowed in math. So, something like six divided by four is fine, but six divided by zero is not. Remember, a rational number must follow the rule of being a ratio of two whole numbers.

In short, rational numbers are all the numbers that can be written neatly as a fraction. They include positive numbers, negative numbers, whole numbers, and decimals that stop or repeat. Rational numbers are very common and easy to work with in math. You will see them in almost every part of your schoolwork. Once you know how to spot them, they are easy to understand.