MA10103: Foundation Mathematics I. Lecture Notes Week 1
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1 MA00: Foundation Mathematics I Lecture Notes Week Numbers The naturals are the nonnegative whole numbers, i.e., 0,,,, 4,.... The set of naturals is denoted by N. Warning: Sometimes only the positive integers (i.e.,,,, 4,...) are called natural numbers The integers are the whole numbers i.e....,, 4,,, 0,,,,.... The set of integers is denoted by Z. The rationals are the fractions, e.g.,,, Fraction = etc. In general, numerator denominator where numerator and a denominator are both integers (the denominator is never 0 and often chosen > 0). The set of rationals is denoted by Q. We simplify by cancelling any common factors, e.g., 6 =. All integers are rational, e.g., =. Note: Adding/multiplying naturals/integers/rationals yields a natural/integer/rational; substracting integers/rationals yields an integer/rational; dividing rationals yields a rational. Multiplying Fractions e.g.: Dividing by Fractions Turn divisor upside down then multiply 7 = 7 = 0 e.g.: 7 = 7 = 4 Adding/subtracting fractions Easy case, e.g.: + = + = 4 or 7 = 7 = = 7 = 4 Less easy, e.g.: + To do this, make a common denominator : + ( multiply by = ) ( + ) = + 0 =
2 Worked Eamples (not given in the lecture) = = = 7 6 8= 9 = 9 = 8 = = 9 8 = Real numbers This is the set of all possible numbers, containing integers and rationals, but also numbers like, π and 7. The set of real numbers is denoted by R. We may show net week that is not a rational number. We can write all reals as decimals, e.g., = 0. (terminates) 4 = = 0. (recurs) = (neither terminates nor recurs) 4 = = 0. (recurs) The line above digits indicate that they recur. Note: Decimal digits of rational numbers either terminate or recur. Decimal digits of irrational numbers neither terminate nor recur. On the all possible numbers (neither given in lecture and nor eaminable) What does all possible mean? Look at the following sequence of rational numbers:,, 7, , ,.... The (n + )th number n+ in this sequence is calculated from its predecessor, the nth number n, by n+ = n + n. E.g., using = one calculates ( + )/( ) = ( 7 4 Now, look at the following table: n n n (6 d.p.) n n (6 d.p.) )/ = 7 =.
3 The squares n approach, and in fact, the sequence of rational numbers n converges to the limit. All possible means that each real number can be obtained as limit of a converging sequence of rational numbers. Note, that the decimal epansion itself yields a converging sequence of rational numbers, e.g.,, 4 0, 4 00, ,... also converges to. Also note that the limit of a converging sequence of real numbers is again a real number. Net week, we might show that is irrational, i.e., a real number that is not rational. We may approimate decimals to a number of decimal places (d.p.) or significant figures (s.f.). Decimal places = st d.p. 7 th d.p. so =.4 ( d.p.), =.446 (7 d.p.) The first decimal place is the first digit after the decimal point, etc. (If the first digit not written is we round the previous digit up). Significant figures = st s.f. th s.f. so = ( s.f.), =.446 (8 s.f.) The first significant figure is the first non-zero digit, regardless of the position of the decimal point. Eamples: 999 = 000 ( s.f.); 999 = ( d.p.) END OF LECTURE
4 Standard Inde form The gravitational constant G is 0. 0 }....{{ } m kg s. 0 zeros It is very inconvenient to write like this, so we use S.I. form : G = m kg s The general form is a 0 b where a < 0 and b is an integer. Eamples: 0.00 = = = ; 999 = 0 ( s.f.) More Eamples. Write the speed of light correct to s.f. c = m s ( s.f.). Write the speed of light correct to s.f. in S.I. form. c = m s ( s.f.). Write Avogadro s constant correct to s.f. in S.I. form. N A = mol ( s.f.) 4. Write π correct to d.p. in S.I. form. π =.4 ( d.p.) Speed of light c = m s ; Avogadro s Constant N A = mol Basic Algebra Multiplication: means multiplied by. Use of brackets: p means p p, while (p) = p p = 9p. Eample: (a) = a a a = a a a = 8a Division Eamples: y = = y = y = y = Of course, we are assuming here that all denominators, e.g., in the first eample, are not zero, because otherwise the left-hand sides would not be defined. To be more precise, the first eample should actually read: equals for all ecept for = 0 for which is not defined.
5 Addition/Subtraction Eample: y + y + y + y st term rd term Like terms contain the same combination of letters, e.g. combined. y and y and can be Unlike terms cannot be combined Eamples: (a) y + y + y + y = 8 y + y + y (b) p q (p) q + pq = p q 4p q + pq = p q + pq We say that pq is the pq term in (b). Its coefficient is Epanding brackets: (a + b) = a + b, (a + b) = a + b = a + b (same as (a + b)) (a + b)(p + q) = a(p + q) + b(p + q) = ap + aq + bp + bq (Leave out the middle step if you like.) More Eamples ( ) = 6 ( )(y ) = y y + ( a)( + a) = a + a a = a A very important formula: ( a)( + a) = a The left-hand side is the factorisation of the difference of two squares. Also: ( + a) = + a + a and ( a) = a + a More generally, for any a and b, ( + a)( + b) = + (a + b) + ab. We can use this to factorise a quadratic epression (i.e., one involving and ) into linear factors (invloving only ).
6 Eample: Factorise Here ab = 6 and a + b =. From the latter we deduce b = a and obtain the following table: a b = a a b Therefore, we obtain a = and b = (or vice versa) and have the following factorisation: = ( + )( + ). You don t have to do such a table! We also note the following: The product of two positive numbers is positive. The sum of two positive numbers is positive. The product of two negative numbers is positive. The sum of two negative numbers is negative. The product of a positive and a negative number is negative. With this, we can analyse the following eamples. Eamples: Factorise Here, we have ab = and a + b = 7, thus both a and b are positive. Indeed, one obtains = ( + )( + 4). Factorise + 4. Here, we have ab = and a + b = 4, thus one of the two numbers is positive the other negative. One obtains + 4 = ( )( + 6). END OF LECTURE
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