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Exploring real numbers
Integers and rational numbers
Natural numbers are all numbers 1, 2, 3, 4… They are the numbers you usually
count and they will continue on into infinity
...
g
...
g
...
A rational number is a number
𝑎
𝑏
,𝑏 ≠ 0
Where a and b are both integers
...
As it can be written without
a decimal component it belongs to the integers
...
2
Or repeating as in
5
= 8
...
If you look at a numeral line
ou notice that all integers, as well as all rational numbers, are at a specific
distance from 0
...
It is shown with the symbol
|x|
If two numbers are at the same distance from 0 as in the case of 10 and -10 they are
called opposites
...
|10|=10=|−10|
Calculating with real numbers
Addition
When adding real numbers with the same sign the sum will have the same sign as
the numbers added
...
The sum will then have the same sign as the number with
the greater absolute value
...
a+b=b+a
And the additive associative property tells us that it also that the order in which we
group three or more numbers does not affect the sum
...
a+0=a
And the additive inverse property tells us that if you add a number with its opposite
you will always get 0
a+(−a)=0
Subtraction
You know by now that
10−2=10+(−2)=8
This means that subtracting 2 from 10 is the same as adding negative 2 to 10
...
a−b=a+(−b)
Multiplication
If you remember from pre-algebra the product of two real numbers with the same
sign is always positive
3 x 2=6
(−5) x (−3)=15
This also holds true if you multiply more than two numbers
...
The product is not affected by order in which you multiply the numbers
a x b=b x a
The same goes for the multiplicative associative property
...
a x (b x c)=(a x b) x c
The multiplicative identity property tells us that if we multiply a number with 1 the
product is always the number
a x 1=a
The multiplicative identity of 0 tells us that the product is always 0 when you multiply
a number with 0
a x 0=0
The last property of multiplication is the multiplicative property of -1 and this property
tells us that a product of a number and -1 is the opposite of the number
a x (−1)=(−a)
Division
Reciprocals are numbers that when multiplied have the product 1
...
Zero does not have a multiplicative
inverse since everything multiplied with 0 is 0 as we could see above
...
The quotient of two numbers with the same sign is positive
8
=4
2
Whereas the quotient of two numbers with different signs is negative
−8
= −4
2
The quotient of 0 and any nonzero real number is always 0
0
=0
1000
Since 0 does not have a multiplicative inverse you cannot divide a number by 0
...
He can calculate this in two different ways
...
Rectangle1: 3 x 5=15
Rectangle2: 3 x 7=21
15+21=36
Or as the height is the same in both rectangles we can multiply the the height with the
sum of the bases
3(5+7)=3 x 12=36
As we can see
3 x 5+3 x 7=3(5+7)=36
This is an example of the distributive property which can be used to find the product of a
number and a sum or a difference
...
A term could either be a number, a
variable, or a product
...
A term that only
contains a number and no variable part is called a constant term
...
The two other terms have the coefficients 3 and 7
...
The constant terms are like terms as
well
...
An expression is written in its simplest form when it contains no like terms and no
parentheses
Example
Simplify the expression
2(3𝑝 + 5) − (𝑝 + 2)
Notice that the second parenthesis is multiplied by -1
...
Square roots
In the first section of Algebra 1 we learned that
32 = 3 x 3 = 9
We said that 9 was the square of 3
...
All positive real numbers has two square roots, one positive square root and one
negative square root
...
The reason that we have two square roots is exemplified above
...
√𝑎
To indicate that we want both the positive and the negative square root of a radicand we
put the symbol ± (read as plus minus) in front of the root
...
√0 = 0
Negative numbers doesn't have real square roots since a square is either positive or
0
...
For example 25 is a perfect square since
±√25 = ±5
If the radicand is not a perfect square i
...
the square root is not a whole number than
you have to approximate the square root (unless we can keep √3 as an answer)
...
73205 … ≈ ±1
...
This means that they can't be written as the quotient of two integers
...
The irrational
numbers together with the rational numbers constitutes the real numbers