TOK(Theory of Knowledge) - Summary of Number

 

Number

In this Chapter, we are going to recap the basic knowledge regarding “Numbers”. The following descriptions are summarized for preparation of past paper practice and if you do not know something, you can contact us.

 

Arithmetic 

·       Decimals         -symbol (.) -expressed to write number values between integers.

·       Fraction          -symbol $\frac{a}{b}$ -can be added or subtracted directly when they have  common denominator.

·       fractions and decimals -A decimal is simply the fraction expressed in tenths, hundreds, etc

 

eg. To convert 2.5 to fraction, it would be $\frac{2.5}{1}$. Then multiply with 10 to both denominator and numerator to get the unchanged value, i.e $\frac{2.5 \times 10}{1 \times 10}$ getting $\frac{25}{10}$. The simplest form will be $\frac 52$

$2.5=\frac{2.5}{1}=\frac {2.5 \times 10}{1 \times 10}=\frac{25}{10}= \frac 52$

 

If the number to convert is 2.52, the multiply number would be 100. (If 2.523 -1000)

 

 

Number Facts and sequences

Number Facts

·       Integers -  positive, zero, negative whole number

·       Prime Numbers – only divisible by 1 and itself and greater than 1.

·       Multiples – eg multiples of 5 are 5,10,15,20

·       Factors – eg factors of 20 are 1,20, 2,10, 4,5

·       Square number – eg $5^2=5\times5=25$, $6^2=6\times6=36$

·       Cube Numbers – eg $5^3=5\times5\times5=125$, $6^3=6\times6\times6=216$

Rational and Irrational Numbers

·       Rational number – can be written as $\frac ab$ where a and b are integers, and $b \neq 0$

·       Irrational number – cannot be represented as simple fractions, eg $(\sqrt 2, \pi) $

Sequences, the nth term

 

Approximation and Estimation

Significant Number

1.     All non zero numbers (345 has THREE significant numbers)

2.     Zero between significant numbers (305 has THREE significant numbers)

3.     Zeros right to decimal (102.00 has FIVE significant numbers)

4.     Trailing zeros in a whole number with decimal (100. Has THREE significant number)

 

Non-Significant Number

1.     Leading Zeros (0.000034 has TWO significant number)

2.     Trailing zeros in a whole number without decimal (100 has ONE significant number)

 

 

Approximation

1. If it is 5 or more, increase the digit by one
2. if it is less than 5, keep digit the same

Eg: Round 5846 to the nearest 10, 100 and 1,000.

·        5846 to the nearest 10 is 5850 (6 is more than 5, the 10^th place 4 is increased by 1)

·        5846 to the nearest 100 is 5800 (4 is less than 5, the 100^th place 8 remains the same)

·        5846 to the nearest 1000 is 6000 (8 is more than 5, the 1000^th place 5 is increased by 1)

 

Standard Form – symbol $(a \times 10^n)$, eg $(2 \times 10^5)$ instead of 200000.

 

Ratio and Proportion

Ratio - Used to describe a fraction, usually to compare values.

Proportion - related problems are usually solved by finding a unit quantity.

 

Foreign exchange – a unit quantity is given and converted to other unit

Eg,       1dollar=1.2 AUD, 10dollar = 12 AUD

 1.2 AUD = 1dollar, 

$1 \times 1.2 $ AUD=1 dollar, 1 AUD = $\frac{1}{1.2}$ dollar

Map Scales – proportion is used

 

Percentage

Convenient way of expressing fractions.

Eg 25% of $200$ = $\frac {25}{100} \times 200$= $\frac 14$ of $200$

[Easy tip – to convert % to fraction, replace % with $\times \frac{1}{100}$   ]

Percentage increase or decrease

Percentage Profit = $\frac{Actual Profit}{Original Price}\times 100$

Percentage Loss = $\frac{Actual Loss}{Original Price}\times 100$

 

 

Simple interest

$I=\frac{P \times R \times T}{100}$,

Where P= invested money, R= percent interest per year, T = time of investment in year, I=interest money

 

Eg, if 12000$ is invested with 5% interest per year for 2 years,

Interest = $\frac{12000 \times 5 \times 2}{100}$= $1200$ $

[Easy tip : Interest = $\frac {PaRTy}{100}$]

 

Compound interest        

$CI=P(1+\frac rn)^{nt}-P$

Where, CI = compound interest, P= Initial amount, r= rate of interest, n= number of times interest is compounded per year, t= time in year

If the interest is compounded annually(per year), the formula derived would be

$CI=P(1+\frac {R}{100})^{t}-P$

 

 

Speed, Distance and Time

Distance = Speed x Time

[s=vt, where: s = distance, v= velocity, t = time]

Disclaimer – This article’s purpose is to recap the Chapter before practicing topic past paper, so that the terms and formulas can be remembered on reading the questions. After this article, you can practice the TOPIC by TOPIC past papers via the following link.

Click here TOPIC by TOPIC Questions - E math 0580 (2017-2018) 

Congrarulations! You are ready to practice past papers. you can click Home button and choose the post you would like to practice. See you in the next chapter.