basic scientific maths

7/29/2019 Basic Scientific Maths

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BASIC SCIENTIFIC MATHS

EXPONENTS & SCIENTIFIC NOTATION

Exponents

Exponent - shows that a number is to multiplied by itself a certain number of times:

e.g. 24 = 2 x 2 x 2 x 2 = 16

102 = 10 x 10 = 100

45 = 4 x 4 x 4 x 4 x 4 = 1024

Thus, the BASE is the number that is multiplied and the EXPONENT is the power to which

the base is raised.:

e.g. 102 - 10 is the base number

2 is the exponent.

When the exponent is a negative number, the reciprocal of the number (1/n) should be

multiplied times itself:

e.g. 10-4 = 1/10 x 1/10 x 1/10 x 1/10

= 1/10000 = 0.0001

When multiplying two numbers with exponents, the exponents should be added.

Thus: am x an = a(m+n)

e.g. 23 x 22 = 25 = 32

105 x 10-3 = 102 = 100

NB. The numbers must have the same base!

When dividing, the exponents are subtracted.

Thus: am/an = a(m-n)

e.g. 43/46 = 4(3 - 6) = 4-3 = 0.015625

10-6/10-4 = 10(-6)-(-4) = 10-2 = 0.01


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Again, this applies only when the base numbers are the same.

If the base numbers are different, the numbers with exponents must be converted to their

corresponding values without exponents and the calculation then performed:

e.g. 45 x 10-3 = 1024 x 0.001 = 1.024

34/62 = 81/36 = 2.25

Again, when adding or subtracting numbers with exponents, the numbers should first be

converted to their corresponding values without exponents, whether the base numbers are the

same or not:

e.g. 102 + 106 = 100 + 1000000 = 1000100

102 - 63 = 100 - 216 = -116

Any number to the power 0 is 1:

e.g. 360 = 1 100 = 1

Exponents of the base number 10:

For numbers > 1:

1. The exponent is the number of places after the number and before the decimal point.

100000.0 = 105

2. The exponent is positive.

3. The larger the positive exponent the larger the number.

105 > 103

100000 > 1000

For numbers < 1:1. The exponent is the number of decimal places to the right of the number including the first

non-zero digit.

0.0001 = 10-4

2. The exponent is negative.

3. The larger the negative exponent, the smaller the the number.

10-2 > 10-5

0.01 > 0.00001


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Thus,

1,000,000 106 (6 places before decimal point)

100,000 105 (5 places before decimal point)



100 102 (2 places before decimal point)

10 101 (1 place before decimal point)

1 100 (no places before decimal point)

0.1 10-1 (1 place right of the decimal point)

0.01 10-2 (2 places right of the decimal point)





Scientific Notation

Uses exponents to simplify handling of very large or very small numbers:

e.g. 602000000000000000000000.0

6.02 x 1023

Thus scientific notation is a number between 1 - 10 multiplied by 10 raised to a power.

A number written in scientific notation has 2 parts:

a) coefficient 6.02

b) exponential 1023

A negative exponent is used for a number less than 1:

0.00000001 = 1 x 10-8

LOGARITHMS

Common Logarithms

Also called logs or log10.

Log10 of a number is the power to which 10 must be raised to get that number:

e.g. 1000 = 103

thus, the log10 of 1000 is 3

0.01 = 10-2

thus, the log of 0.01 is -2.


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Thus, the logs of numbers 2, 3, 4, 5, 6, 7, 8 and 9 are decimals between 0 and 1 (log of 1 = 0

and the log of 10 = 1).

Logs of numbers between 0 and 1 have negative values (log10 of 1 = 0). Log10 of 0.5 = -

0.301.

Log10 values are found using log tables or a scientific calculator.

Common logs of powers of ten:

1,000,000 = 106 log 106 = 6

100,000 = 105 log 105 = 5

10,000 = 104 log 104 = 4

1,000 = 103 log 103 = 3

100 = 102 log 102 = 2

10 = 101

log 101

= 11 = 100 log 100 = 0

0.1 = 10-1 log 10-1 = -1

0.01 = 10-2 log 10-2 = -2

0.001 = 10-3 log 10-3 = -3

0.0001 = 10-4 log 10-4 = -4

0.00001 = 10-5 log 10-5 = -5

0.000001 = 10-6 log 10-6 = -6

The antilogarithm is the number corresponding to a given logarithm:

e.g. antilog of 3 = 1000

antilog of 0.845 = 7

Natural logarithms (ln or loge) have a base number of 2.7183. These are not generally used in

biomedical laboratories.

The terms log and ln are not interchangable.

Applications of Logarithms e.g. pH

pH is used to express the concentration of hydrogen ions or [H+] in a solution.

The pH of solutions is very important when performing analytical tests in the biomedicallaboratory.

The [H+] in solutions normally varies from 0.00000000000001 (or 1 x 10-14) to 1.0 M .

Soren Sorenson devised the pH scale where the pH of a solution is equal to the negative log

of the hydrogen ion concentration:

pH = - log [H+]

Thus, if [H+] = 1 x 10-14, the pH = 14 and

if [H+] = 1 M, the pH = 0.

If the pH of a solution is 5.60, the [H+] is the antilog of -5.60, i.e. 2.51 x 10-6 M.

basic scientific maths

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