2 intro to mathematics

7/28/2019 2 Intro to Mathematics

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INTRODUCTION

TO MATHEMATICS


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Introduction

Transformation of science was

made in the 17th century when it

was learned that it can be

expressed mathematically

Ideas expressed in numbers are

said to be unambiguous.

It does not give double meanings.


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Introduction

Findings expressed mathematically

are easier to verify or to disprove

by experiment.

The use of mathematics gave way

to the enormous success of science.


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SIGNIFICANT FIGURES

It is the number of reliably known

digits a numerical quantity

contains.

For measured quantity, usually it is

all the digits that can be read

directly from the instrument used

in making the measurement.


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RULES: NON-ZERO

All non-zero digits are always

significant.

Ex. 1, 2, 3, 4, 5, 6, 7, 8, & 9


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RULES: The Digit Zero

Zeros may not be significant,

depending on whether they mark

the decimal point or indicate a

measured value.

Leading Zeros

Confined Zeros

Trailing Zeros


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RULES: LEADING ZERO

Zeros located at the beginning of a

number are NEVER significant.

They merely locate the decimal

point.

Ex.: 0.0897 3 SF


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RULES: CAPTIVE ZERO

Zeros located between non-zero

digits are ALWAYS significant.

Ex.: 2805.3 5 SF


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RULES: TRAILING ZERO

Zeros located at the end of a

number ARE SIGNIFICANT ONLY if

the number has an EXPLICITLY

SHOWN DECIMAL.

Ex.: 123.00 5 SF


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In WHOLE NUMBERS WITHOUT A

DECIMAL POINT that end in one or

more zeros, the zeros may NOT BE

SIGNIFICANT.

Ex.: 3900 2 SF


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Numbers expressed in scientific

notation with zero, follows the rule

in decimal number.

Ex.: 3.0 x 102 2 SF


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EXERCISES

1. 25.25

2. 200.5

3. 0.0025

4. 0.025

5. 300

6. 0.00500600701007. 5.890 x 105

8. 0.00234 x 10-4


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ROUNDING OFF

NUMBERS

If the next digit after the last

significant figure is 5 or greater,

round up.

Increase the last significant figure

by 1.

Example: 2.136 becomes 2.14

rounded to 3 SF


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ROUNDING OFF

NUMBERS

If the next digit after the last

significant figure is less than 5,

round down.

Do not change the last significant

figure.

Example: 2.132 becomes 2.13

rounded to 3 SF


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EXERCISES

1. 5679

2. 986.981

3. 0.087624. 0.0123

5. 45.81


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Significant Figures in

Calculations

For addition and subtraction:

Round off the sum or difference

based on the least number of digits

after the decimal point.


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Significant Figures in

Calculations

For multiplication and division:

Round off the product or quotient

based on the number that contains

the least number of SF.


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SCIENTIFIC NOTATIONS

It is a shortcut way of expressing

very large and small numbers.

Numbers are expressed into

greater than but less than ten with

a power of ten.


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RULES: Scientific Notation

To express a number greater than 1

in scientific notation, we count the

number of places the decimal point

has to be moved to the left to putjust after the first digit of the

number.

The number of movement ofdecimal point equals the positive

exponent.


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EXAMPLE

30,000,000

can be expressed as3 x 107


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RULES: Scientific Notation

To express a number smaller than 1

in scientific notation, we count the

number of places the decimal point

should be placed after the firstnon-zero digit.

The number of movement of

decimal point to the right equalsthe negative exponent.


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EXAMPLE

0.0008

can be expressed as

8 x 10-4


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EXERCISES

1. 7,020,000

2. 847

3. 0.0004624. 0.00000184

5. 72.4


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RULES:

Multiplication and Division

In multiplication, the numerical

parts are simply multiplied

together and the exponents are

added.

In division, the numbers are

divided and the exponents are

subtracted algebraically.


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EXAMPLE

(2 x 103) (2 x 102)

= (2 x 2) x (103+2)

= 4 x 105


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RULES:

Addition and Subtraction

In adding or subtracting numbers

in scientific notation, the exponents

must be the same number


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EXAMPLE

(2.5 x 102) + (1.20 x 102)

= (2.5 + 1.20) x (102)

= 3.7 x 102


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EXAMPLE

(2.5 x 102) + (1.20 x 103)

= (2.5 x 102) + (12.0 x 102)

= 14.5 x 102

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