## Power and roots

Writing repeated factors can be so tiring! Powers and exponents are a shorthand notation that save space and are a quick and easy way to simplify multiplication

Let’s get started

Exponent – A number that shows how many times another number is multiplied times itself

3^2 = 3 x 3 = 9

Evaluate means to find the value of something.

Example Evaluate 2^3 x 5^2

Here’s how

2^3 x 5^2 = 2 x 2 x 2 x 5 x 5 = 200
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## Multiples and factors

Do you draw a blank when you see LCM and GCF? Don’t worry, these acronyms stand for concepts that make sense. They help you find the common denominator of two or more fractions, simplify fractions, and add and subtract fractions.

Let’s get started

Multiple – The product of a given number and a nonzero whole number.

Do you draw a blank when you see LCM and GCF? Don’t worry, these acronyms stand for concepts that make sense. They help you find the common denominator of you or more fractions, simplify fractions, and add and subtract fractions.

Let’s get started

Multiple – The product of a given number and a nonzero whole number.

Multiples of 2 are

2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6, and so on.

Example Find the first four multiples of 8

## Properties of Operations

You probably use properties of operations every day without even giving them a thought. You may have noticed that 3 x 4 and 4 x 3 are both 12. That”s an example of the commutative property of multiplication. The ideas are probably familiar – you just need to brush up on the vocabulary.

Let”s get started

The associative and commutative properties hold for both addition and multiplication.
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## Prime and composite numbers

Why on earth do you need  to know about prime and composite numbers? Fractions! Work with fractions becomes much easier if you recognize prime and composite numbers. It helps you know if a fraction needs to be simplified.

Let’s Get Started

Here are some of the terms you will encounter when working with numbers.

Factor – A number that is multiplied by another number to find a product

2 and 3 are factors of 6, because 2×3 = 6.

Prime number – A number with only two factors, itself and 1.

7 is prime because its only factor are 1 and 7.
1×7 = 7

Composite number – A number with more than two factors.

6 is a composite number because 1, 2, 3, and 6 are all factor of 6.
1 x 6 = 6        2 x 3 = 6

In the information society. communication is everything, In order to communicate information, we have to speak the same language and use the same vocabulary.

Let’s Get Started

Our number system in a base -10 place-value system. We use 10 different digits to form all the numbers. The value of the digit depends on which place it is in.

Period – A grouping of three palces in the place-value chart. Each period has a ones place (O), a tens place(T), and a hundreds place (H).

Standard Form – A number written with digits.
124
614,900
70,142, 859

Word form – A number written with words.

One hundred twenty-four
Six hundred fourteen thousand, nine hundred
Seventy million, one hundred forty-two thousand, eight hundred fifty-nine.

Expanded form –  A number written as the sum of its place values.

124 = 100 + 20 + 4
614,900 = 600,000 + 10,000 + 4,000 + 900
70,142, 859 = 70, 000, 000 + 100,000 + 40,000 + 2,000 + 800 + 50 + 9

Sometimes expanded form is written this way to show the place values even more clearly:
614,900 = 6 x 100,000 + 1 x 10,000 + 4 x 1,000 + 9 x 100

about expanded form – it doesn’t come up very often in everyday life. Expanded form helps kids understand the amazing idea of place value and the role of zero as a placeholder.

Now. Let’s practice

Write each number in word form.
1. 3,004                 2. 5,700,054            3. 37,000,416

Write each number in standard form.
4. Seventy-five thousand        5. sixteen million, thirty-four

Write each number in expanded form.
6. 45,700     7. 190, 063            8 5,009,802

## Numbers: The basics

A number is a number, right?? Well, not exactly. There are different types of numbers. Kind of like a dog and a cat are both animals, but they are different types of animals.

Numbers and number systems are pretty basic in math, so let’s look at some definitions and think about why we need all the different types.

Let’s  Get Started

The numbers you use all the time are called the counting numbers.

Counting numbers – The most familiar and basic numbers. The numbers we counts with: 1,2,3,4,5,6…..

When we throw in 0, we have a whole new set.

Whole numbers – The counting numbers are zero: 0,1,2,3,4,5,6…….

Whole numbers were all that we needed for awhile – until we needed a way to talk about loss of money or yards in a football game or depths below sea level.

Integers – The whole numbers and their opposites.

An integer that is divisible by 2 is called an even number. If there is a remainder of 1 when the integer is divided by 2, the integer is called an odd number.

But what about when you share a cookie with your friends? You need a fraction to tell how much cookie you each get to eat. This brings us to the rational numbers.

Rational numbers – Numbers that can be written as a ratio of two integers. Fractions and decimals are ratios of integers. All the integers are rational numbers.

Irrational numbers – Numbers that cannot be written as the ratio of two integers.

Real Numbers – The rational numbers and the irrational numbers.

## Introduction of Hindu-Arabic & Roman numerals

At first, the ancients developed names for the numbers. They spoke of having one sheep, two sheep, etc. But you can see how difficult it would be to add or subtract columns of numbers expressed only in words. Thus we learn that arithmetic computation did not begin until man came to use symbols for numbers. The kinds of symbols used for numbers went through various changes starting with the simple vertical mark of ancient Mesopotamia, progressing to the combination of the Egyptians, the familiar numerals of the Romans, and finally to our present figures.

We are indebted to the Arabs for our present method of writing numbers. For this reason, the numerals 0 through 9, the ingredients for any number combinations we wish to write, were called Arabic numbers for a long time. But more recently historians have discovered that the system of writing numbers now used by civilized people throughout the world was originated by the Hindus in India. The Arabs learned the system from the Hindus and are credited with having brought it to Europe soon after the conquest of Spain in the eighth century AD. For this reason, we now property call it the Hindu-Arabic system of numerals.

An early system of writing numbers is the Roman system. It is generally agreed that it is of little practical value in today’s world of advanced mathematics.

Because you will still see Roman numerals used in recording dates, in books, as numbers of a clock face, and in other places, it is worth taking a little time to learn how to read them.

The Roman number system is based on seven letters, all of which are assigned specific values. They are:

I=1, V=5, X=10, L=50, C=100, D=500, M=1000

Rule 1: When a letter is repeated, its value is repeated.
eg, I=1, II=2, III=3, XX=20, CCC=300

Rule 2: When a letter follows a letter of greater value, its value is added to the greater value.
eg, VI=6, XV=15, LX=60, DC=600

In these examples, observe that the smaller value I after V means add 1 to the 5 to give 6. In the same way, the V following the X means add 5 to 10 which equals 15. Similarly, LX represents 10 added to 50 to give 60. To write 70, merely add XX after the L to give LXX. In like manner, to write 800, add CC after DC to give DCCC.

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## Some basic ideas of arithmetic

Vocabulary
There are four basic operations in mathematics: addition, subtraction, multiplication, and division. Often when we talk about a collection of numbers, such as the numbers 1, 2, and 3, we use the word set. We could use set notation with braces, [ ], to list the number: [1, 2, 3]. The set of even numbers could be written as [2, 4, 6, 8, 10, …], and the set of odd numbers as [1, 3, 5, 7,…]. (The three dots indicate that the numbers continue indefinitely. In any collection of numbers ending in dots, there is no largest number.)

Here, we deal with two sets of numbers: the counting numbers [1, 2, 3, 4,…] and the whole numbers [0, 1, 2, 3,…]. The whole numbers are just the counting numbers plus zero. When we count, we start with 1. When we answer the question “How many?” we need zero as a possible answer.

Symbols are necessary to make mathematical statements complete. For example, we use symbols for addition (+) and multiplication (X).

= as in 8 + 3 = 11
8 plus 3 equals 11

< as in 3 < 8
3 is less than 8

> as in 8 > 3
8 is greater than 3

Notice that the symbols for less than and greater than are always open toward the larger number. When statements are not true, we put a slash through the symbol:

6 + 3 =/ 11
6 + 3 does not equal 11

5 >/ 7
5 is not greater than 7

9 </ 6
9 is not less than 6

Numerals are symbols for numbers, which are abstract ideas. For example, a fisherman 8000 years ago might record that he caught ||| fish. We could write 3 for the amount ||| and 3 are the symbols for the same numbers. Our number symbols are called arabic numerals.

Digits are the number symbols (numerals) 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in our number system. Numbers are written as combinations of any of these ten digits.

A whole number is written as a string of digits, 7 is a one-digit number; 32 is a two-digit number with 3 as the first digit and 2 as the second digit; 487 is a three-digit number with 4 as the first digit, 8 as the second digit, and 7 as the third digit.

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