The binary number system

Previously we we learned about place value. The one that made me interesting was each place value was increasing by the power of 10.


  For example, 456015 = (4 x 10 ⁵) + (5 x 10 ⁴) + (6 x 10^3) + (0 x 10^2) + (1 x 10^1) + ( 5 x 1)
                                    = 400000 + 50000 + 6000 + 0 + 10 + 5
                                  

Average

 In this lesson, we're going to learn about average. Average means the median (middle score) in sets of scores.
Therefore it's not difficult to see that average of  two scores 10 and 14 is 12. Another way of figuring out is add two score and divide by the number of scores.

The first example is, find the average of 6, 9, 2 and 7.

           Average= 6+9+2+7/4                                                      Find the total of 4 scores and
          =24/4                                                                               divide total by 4.
          =6 

 The second example is Jimmy wrote 5 numbers on a separate paper, the scores are 56, 45,75, 23 and 77.
   
            Average = 56+45+75+23+77/5
                         = 276/5                                                         Find the total of 5 scores
                         = 55.2                                                          and divide the total by 5.

 The third example is The average of 6 number is 54. A fifth number is added onto the total and the new average is 95. What is the fifth number?

           Total of first 4 numbers = 4 x average = 6 x54 = 324
           Total of 5 numbers = 5 x new average = 5 x 95 = 475
           The fifth number is therefore 475 - 324 = 151.


Fourth and last example is, My average in 5 maths test is 85%. If I scored 96% in my sixth test, what is my new average?


                      Total marks in first 5 tests = 5 x average = 5 x 85 = 425
                      Total marks in six tests = 425 + new mark = 425 + 96 = 521

                      New average = 521/6
                                           = 86.8%
                                           =87%  (To the nearest whole number)

                 

Maths signs into words

 In this lesson we're going to learn the maths operation signs and changing it to words. in most of the maths exams, the questions are all about reading and comprehension and most of the words are connected to maths.

  These are the important words tat you need to know.

Sum means add, it means finding the total amount.           The sum of 12 and 21 is 12+21=33

Difference means subtract.                                              The difference between 9 and 3 is 9-3=6

Product means multiply.                                                   The product of 4 and 5 is 4x5=20

Dividend is divided by the divisor                                      20 ÷ 4=5
to give the answer as a                                                         20: dividend
quotient.                                                                             4: divisor
                                                                                           5: quotient

Increase means to add on.                                               Increase 14 by 9 means 14=9=23

Decrease means to subtract form.                                  Decrease 56 by 34 means 56-34=22.

= means equals.                                                                 4+5=9

≠ means not equal to.                                                         4+6 ≠12


> greater than.                                                                   12>5

≥ Greater than or equal to.                                                 X ≥7 means X=7,8 or 9 etc.

< means less than.                                                              4<5

≤ means less than or equal to.                                             X ≤ 14 means X=14, 13, 12 or 11 etc.


∴ means therefore.                                                            X+4=12       ∴  X=8

i.e. means that is or therefore.                                              X=12+7+3 i.e. X=22

≈ means approximately or equal to.                                     The price of the chocolate
                                                                                           is $2.95we can say the cost ≈ $3.

Squared means power of 2.                                                       Five squared means 5 ^2=25

√ means square root.                                                         √16=4 what number does it times itself to get                                                                                                        16?

Cubed means power of 3.                                                  5 cubed 3= 5^3=125.

3^√ means cube root.                                                        3^√125=5. What number multiplied by itself                                                                                          by 3times is equal to 125?  5x5x5=125

Average means middle.                                                      The average of 4 and 10 equals 7.

Evaluate means work out the answer.                                 Evaluate 7x5+1
                                                                                         Answer: 36.

Simplify means find the simplest answer.                             Simplify 8+12-5                             
                                                                                         answer = 15.

* The meaning of the simplify is the same meaning as the word evaluate.

Brackets or grouping symbols

 Last lesson we looked order of operation. This lesson we're going to learn about Brackets or grouping symbols.

There are 3 types of brackets in maths, Parentheses (  ), Brackets [  ] and Braces { }. When doing the order of operations, and there are grouping symbols are involved, do the innermost one grouping symbol first then do the next innermost grouping symbol.

In order of operations we learned the steps to doing the brackets and 4 operations, in this exercise we're learning make the order of the operations bit easier.

The first example is 50-{16+(50/5)}.                                 1) Work out (50/5).
                             = 50-{16+10}                                     2) Work out {16+10}.
                             =  50-26                                              3) Do the subtraction.
                             =24

Te second example is 72/ [76-(22+46)].                           1) Work out (22+46).
                                = 72/[76-68]                                    2) Work out [76-68]
                                = 72/8                                              3) Do the division.
                                = 9

 Third and last example is {[95-(3+2)]/5}+36x2.
                                  = {[95-(3+2)]/5}+36x2                1) Evaluate (3+2)=5
                                  = {[95-5]/5}+36x2                       2) Evaluate [95-5]=90
                                  = {90/5}+36x2                             3) Evaluate {90/5}=18              
                                  = 18+36x2                                    4) Evaluate 36x2=72                       
                                  = 18+72                                        5) Evaluate 18+72=90
                                  = 90

* Some of the people are having a trouble of the different way of asking in long questions. In fact, "evaluate, simplify and give the simplest answer for" are all different ways of asking the same questions.

Order of operation

 In this lesson we're going to learn the order of operation. The definition of order of operation is the agreement that mathematicians around the world agreed on a definite order of doing brackets and 4 operation (+ - x /) otherwise confusion would occur.

   The agreed order of operation

1. Work out the brackets first. (   )
2.Secondly, work out any multiplication or division as they occur from left or right.
3. Lastly, work out any addition or substitution as they occur from left or right.

The first example is 4+5x9

Like the agreement of the order of operation said:
Firstly we do the brackets first but there's no brackets,
so we're going to do the multiplication.

5x9=45
Then we do the addition. 45+4=49
Therefore the answer is 49.

The second example is 18-3x4+2

We do the multiplication first,-3x4=-12
secondly, we do addition and subtraction.18-12+2=8
Then our answer is 8.

Third and last example is 6+40/(9+1).
Firstly, we do the brackets first, (9+1)=10.
Secondly, we do multiplication or division, 40/10=4.
Third and last we do the additions, 6+4=10

Then the answer is 10.

Adding, subtracting, multiplying and dividing whole numbers.

This time we will do some few basic skills of mathematics. We're gonna do adding, subtracting, multiplying and dividing this time.

Now first example will be the addition. For example,
659+12346+7548+23.
Now this is how you do, do it like this. In a easy way to do it is make it into two groups and use the brackets, then we have this (659+12346) and (7548+23). Secondly we solve each brackets and we get this, 13005 and 7571. then we add each other and we get 20576.

Second example is subtraction. For example, 9681-4753.

This is how you do.                   9681
                                             -  4753                                               

Now we can't do 1-3 so we borrow 1 from 8 and that makes us 11 and 11-3 equals to 8, ten 8 becomes 7 because he let borrow 1 from him. Therefore is 7 so 7-5 equals to 2. Same as 1-3, we can't do 6-7 so we borrow 1 from 9 and tat makes 16 so 16-7 equals to 9. Lastly 9 becomes 8 and 8-4 equals to 4.

 Therefore we got 4928.

Third example is multiplication. I'll show you 2 examples of the multiplication.
 First is 579x6. For easiest way is doing it like this.      327                                                
                                                                               x     7

Now we don't just multiply the whole number by itself we multiply the each whole number above 6, so we do it like this.

I'll show you a step how to do it. every time  when do the calculation like this, you start from right.
1) 7x7=49
2) Write down 9 and carry over 4. ( Put 4 above 2.)
3) 2x7=14, then we add 4, so is 18.
4) Write down 8 and carry over 1. (Put 1 above 3.)
5) 3x7=21, add 1= 22.
6) Write down 22.

In the end you should get 2289.

 second example is 24x9765. Steps of solving is little bit different to the eg1 multiplication.
1) Multiply top line by 4 in usual way to obtain 39060.
2) Write down a zero under zero or just leave the space open.
3) Multiply the top line by 2 in the usual way to obtain 19530.
4) Now simply add the 2 rows to get the answer 234360.

The working out should be like this.

                                                           9765
                                                          x   24
                                                         39060
                                                       19530
                                                     
                                                     =234360

Forth and last example is division.
 For example, 5126/24. I'll show you the steps how to do it.

1) 51/24=2
2) 2x24=48
3) 51-48=3
4) Bring down the 2
5) 32/24=1
6) 1x24=24
7) 32-24=8
8) bring down the 6.
9) 86/24 = 3
10) 3x24 = 72
11) 86-72 = 14
12) can't do 14/24
13) Record the remainder.

 Your working out should be like this.
                                                                    213  r  14
                                                            24   5126                                                      
                                                                   48
                                                                     32
                                                                     24
                                                                       86

                                                                       72
                                                         
                                                                       14

The four operation

Even if you have your fancy calculator and solves the question fast. It's about understanding and that is understanding from the scratch. The important to understand and memorize in your head is the four operation, addition, subtraction, division and multiplication.

 It is very important that you have to understand and knowledge of the basic skills for rest of your life. In my life, most of my day is maths. Everything is all about mathematics, you can't just get away from mathematics, it follows you all the time.

Although there's lots of formulas to you to memorize, understand and knowledge but these formulas are all about the four operation.

Rounding off

In maths, sometimes we get a complicated number like a whole number then the decimal numbers comes along!
So we round off, this is how you do.

To round off a number,
* We round up (add 1 ) if the next figure is 5 or more.
* We round down (leave it) if the next figure is less than 5.

here's the first example, 5429.
Now we're going to round it of to the nearest ten, now we see that 9 is more than 5 so we round off and add 1 to 2 and put zero where 9 is staying, therefore is 5430.
second, now we're going to round off to the nearest hundred. Now, 429. 2 is smaller than 5, so we change 2 and 9 to zeros.
Third, we're going to round it off to the nearest thousand. 5429, 4 is smaller than 5, so we change 4,2 and 9 to zeros.

Second example is 759345. Firstly, to round it off to the nearest ten, 5 (next to the 4) is 5, so we round it up to 50 and we got 759350. Second is to round it off to the nearest hundred. 345,4 (next to 3) is smaller than 5 so we round it off to 759300.
Third would be rounding off to the thousand. now 9345, we can see that 3 ( next to 9) is smaller than  5 so we change 3,4 and 5 to zeros and we have 759000. Forth is rounding off to the ten thousand. 59345 9 is bigger than 5 but 3 is smaller than 5 so we change 3,4,5 and 9 to zeros and we got 750000.




Now lastly, rounding off is used most in our like rounding the number of people at mall or at some events that you want to call, secondly it uses in counting money to calculate the approximation of the amount of money.

Scientific notation

In Maths we often multiply the same number itself by several times. So to reduce the time of calculation, most of people uses the scientific notation.

For example, lets say we're multiplying 3 for 7 times.
                                         3x3x3x3x3x3x3

Then we do this. 3⁷ 

This is a example of the scientific notation but also it's called the Exponential notation. 

Now, the number 7 above the 3, is called the index. Reason we call index is because we shortened the calculation, so we're showing the proof of the calculation. 

Also, some of the people reads 3 times 3 times 3 3 times... we don't pronounce like that. We pronounce like this three to the power of seven. 

Second example is 25x10⁵. In this case, firstly solve the 10 to the power of 5 then multiply with 25.

Also when the whole number is decimal, for example, 5.345x10⁷. First one is do the 10 to the power of 7 first then multiply with the 5.345. Second one is move the decimal point to the right by the number of index, in this one we move decimal point to the right 7 points then put the zeros.

Third and last example is 4.5x10⁻⁶. This example is similar to the eg1 and eg2 and the solving way is similar but this we're putting the decimal point at left. Reason is the minus point means going backward and that means the answer will be the decimal. Therefore, we send the decimal point to left 6 units away. Then we got the answer 0.0000045.

* Sometimes scientists or Mathematicians uses scientific notation because sometimes they do the complicated mathematics and often they multiply the same numbers several times.