Christian Wilhelm Puritz
I was born in Berlin during WW2, We moved to London in 1948. I studied at Wadham College, Oxford, where I got a 1st in maths in 1962. It was during that time that I became a Bible believing Christian. After various attempts at research, and a year computing at AERE Harwell, I did research at Glasgow University for a Ph D and then taught maths at the Royal Grammar School, High Wycombe, from 1970 to 2001. I wrote Venture Mathematics Worksheets, and compiled and translated "Christ or Hitler?"
Subjects: Education
Biography
I was taken to a German Lutheran church in London in my boyhood, and confirmed at 15, but had never really read the Bible, nor been taught the need for a personal conversion (being born again, as Jesus teaches Nicodemus in John 3.) At Oxford I got into discussion with a fellow undergraduate, Edward Greene (who years later founded what is now Greene's Tutorial College) who knew and believed what the Bible teaches about the seriousness of sin, the holiness of God, and the wonderful sacrifice of Jesus provided in love, at great cost, so that sin can be forgiven. It started me reading the Bible, and for a time I didn't know whether to believe this; for if I did, it meant I was not yet a real Christian at all! But I did become convinced that this way of being saved through the free gift of God could not have been made up by man (who thinks naturally of religion as something we do to get right with God). I came to believe the gospel taught in the Bible, and have been a follower of the Lord Jesus since then.I married Cynthia, who worked in Wycombe :Hospital as a theatre nurse and nurse and ultimately as the sister in charge, in 1986, when we were both in our forties; so we had no children of our own, but quite a lot of input to other children and young people!
Areas of Research / Professional Expertise

My PhD research was in Nonstandard Analysis, a fairly new branch then (1960s) which fascinated me: it was an extension of the number system to include infinitely large and infinitely small numbers, which allows standard mathematical ideas to be defined in the way that they were originally conceived: e.g. that a function is continuous at x=a if an infinitely small change in x results in an infinitely small change in f(x). My research however was to do with infinitely large integers: if m is one, could there be another integer n that is so much greater than m that no function f mapping N to N would make f(m)>n? If so, I said n is in a "higher sky" than m.
After I started school teaching I investigated and wrote articles on subjects relevant to that, including in the early years how school pupils can learn a way to calculate pi, and how they can be helped to derive the formulae for the volume and surface area of a sphere. These are included in my book "Explaining and Exploring Mathematics."
Personal Interests

My wife and I belong to a local Baptist church, and I help out once a week with a book table in town, as well as doing some evangelism door to door. I have written and printed leaflets to help explain the gospel.
I do a fair amount of home tuition, mostly voluntary, particularly with Asian families of whom many have settled in Wycombe, and mostly helping with maths, though i do have one youngster just turned 5 who has learned his letters but still can't get s, i, t to combine to make the word sit!
My book "Christ or Hitler?" contains stories from the life of Wilhelm Busch, who was a youth pastor in Essen from 1930 to 1962, which included the Hitler years, during which he was pursued and several times imprisoned by the Gestapo because of h is insistence on preaching the supremacy of Christ as Lord.
Books
Articles
Using Patterns with Negative Numbers — or, Do Two Minuses Really Make a Plus?
Published: Feb 13, 2017 by Mathematics in School, Vol. 33, No. 4 (Sep., 2004), pp. 2627
Authors: Christian Puritz
Subjects:
Education
Shows by use of patterns (e.g. 73=5, 72=5, 71=6, 70=7, what should 7(1) be?) that adding a negative number is same as taking away a positive no., taking away a negative no. is same as adding a positive one, and multiplying or dividing by a negative no. causes a change of sign. Practical examples are also used: if you have a bank account with 1000 in it, and a kind uncle takes that from you he is giving you 1000. This is included in my book Explaining and Exploring Mathematics.
Use Hundreds and Thousands, Not Apples and Pears! Helping beginners make sense
Published: Jan 01, 2010 by Mathematics in School, Vol. 39, No. 1 (JANUARY 2010), pp. 67
Authors: Christian Puritz
Subjects:
Education
Why does 3a + 4b not equal 7ab? Is it because 3 apples and 4 pears don't make 7 apples pears? But 2a times 3b does = 6ab, so do 2 apples times 3 pears make 6 apples pears? The problem is that a and b don't stand for fruits but for numbers, and 3 1000s plus 4 100s make 3 400, they don't make 7 of anything. The paper continues with more ways of making algebra make sense by relating it to things the pupils already know in arithmetic.
Dividing by Small Numbers — and Why Not by 0?
Published: Nov 01, 2005 by Mathematics in School, Vol. 34, No. 5 (Nov., 2005), pp. 24
Authors: Christian Puritz
Subjects:
Education
Examples like 12*4, 12*2, 12*1 lead to 12*(1/2), 12*(1/4), showing that multiplying by 1/n is dividing by n. dividing by 1/n undoes multiplying by 1/n so is multiplying by n. Dividing by 0 is considered via questions like "I multiplied a number by 0 and got 5; what was the original number ?" With response: there's no such number. When 5 is replaced by 0, the answer is that the original could be any no. Multiplication by 0 can't be undone. This is in my book Explaining and Exploring Maths.
Roots of Integers Are Integers or Irrational, OK?
Published: Sep 01, 2005 by Mathematics in School Vol. 34, No. 4, Sep., 2005
Authors: Christian Puritz
Subjects:
Education
If a fraction a/b is in lowest terms and is not an integer, then powers of a/b are more complicated fractions and cannot be integers. Hence all roots of integers are either integers or are irrational.
Transformed Averages
Published: Nov 01, 2004 by The Mathematical Gazette, Vol. 88, No. 513 (Nov., 2004), pp. 441446
Authors: Christian Puritz
Subjects:
Education
For any function f the ftransformed mean of a set of numbers is made by applying f to the numbers, finding the arithmetic mean (AM) and then applying the inverse of f to the result. The geometric mean (GM) is got by using log as f, while the root mean square uses squaring as f. We ask how the size of the transformed mean compares with the AM; the answer depends on whether f has positive or negative second derivative.
The md Method
Published: May 01, 2004 by Mathematics in School, Vol. 33, No. 3 (May, 2004), pp. 811
Authors: Christian Puritz
Subjects:
Education
Given the sum S of two numbers and their product P, calling the numbers md and m+d leads to 2m=S and m^2d^2=P, whence m and d can be found quite easily. The method also helps with other simultaneous equations such as 4x5y=36, xy=360, by letting u=4x=m+d, v=5y=md, while uv=7200. The method is included in my book on Explaining and Exploring Mathematics.
Negative and Fractional Indices: Making Them Make Sense
Published: Mar 01, 2004 by Mathematics in School, Vol. 33, No. 2 (Mar., 2004), pp. 910
Authors: Christian Puritz
Subjects:
Education
Starts with 1 * 3^4 = 81, 1 * 3^3 = 27, down to 1 * 3^1 = 3. Observe that answers divide by 3 each time, so next lines are...? This leads to definition of a^0 and a^(n). For fractional powers consider 1 * 16^(1/2); can we multiply by 16 in two equal stages? Yes, * by 4. More examples lead to a^(1/n) as nth root of a. This is included in my book Explaining and Exploring Mathematics.
Small percentage changes: why are these not in the syllabus?
Published: Jan 01, 2004 by Mathematics in School; Jan2004, Vol. 33 Issue 1, p6
Authors: Christian Puritz
Subjects:
Education
If u and v change by small percentage amounts, the percentage change in uv is approximately the sum of the individual percentage change, that in u/v is approximately the difference, and hence the percentage change in kx^n is approximately n times the percent change in x. This is applied to approximate calculations, also to the product and quotient rules and the differentiation and integration of x^n.
Calculating π
Published: Nov 01, 2003 by Mathematics in School, Vol. 32, No. 5 (Nov., 2003), p. 36
Authors: Christian Puritz
Subjects:
Education
Start with a 60 degree sector and arc AB of a circle of radius 3; the arc length is pi. The chord AB has length 3. Let M1 be the midpoint of arc AB. Join chords AM1, M1B and calculate the length of each using Pythagoras' then 2AM1 is a better approximation than AB to pi. Bisect the arc M1B at M2 and find chord M2B. 4 times M2B is next approximation, and so on. This approach is simpler than the one in the June 1974 Gazette, and is in my book "Explaining and Exploring Mathematics".
The Difference of Two Squares
Published: Sep 01, 2003 by Mathematics in School, Vol. 32, No. 4 (Sep., 2003), pp. 3234
Authors: Christian Puritz
Subjects:
Education
Starts with class finding (a^2  b^2)/(ab) for various a and b, leading to idea this = a+b. Shown via a square office of side a being partitioned into smaller office of side b + corridor of width ab and length a+b. Result used to do sums like 36^2 34^2, to find sum to n terms of 1+3+5..., to factorise large numbers and to tackle various problems. This is in my book Explaining and Exploring Mathematics.
Extending Pascal's Triangle Upwards
Published: Mar 01, 1981 by The Mathematical Gazette, Vol. 65, No. 431 (Mar., 1981), pp. 4244
Authors: Christian Puritz
Subjects:
Education
The wellknown procedure for extending Pascal's triangle downwards can also be applied in the upward direction, giving coefficients for the powers of x in the expansion of (1+x)^n for negative values of n.
Repaying a Loan by Instalments
Published: Dec 01, 1977 by The Mathematical Gazette, Vol. 61, No. 418 (Dec., 1977), pp. 253261
Authors: Christian Puritz
Subjects:
Education
An amount £A is borrowed with interest rate r% per annum and is repaid in n years by monthly instalments of £I each. This paper derives I in terms of A, r and n using geometric progressions. it is included now in a chapter of my book on Explaining and Exploring Mathematics.
Area and Volume of a Sphere
Published: Oct 01, 1973 by The Mathematical Gazette, Vol. 57, No. 401 (Oct., 1973), pp. 206207
Authors: Christian Puritz
Subjects:
Education
It's about finding the volume and the surface area of a sphere by methods comprehensible to precalculus students, by considering slices of a hemisphere, a cone and a cylinder resting on the same plane (the cone point downward) and showing that hemisphere slice + cone slice = cylinder slice. The area is found by doing a small enlargement of each of the bodies and considering the layers of extra volume. This is included in my book Explaining and Exploring Mathematics.
Ultrafilters and Standard Functions in NonStandard Arithmetic
Published: Jul 01, 1971 by Proc Lond Math Soc Volume s322, Issue 4 Pages 585–768
Authors: Christian Puritz
If m and n are infinite integers in a nonstandard model M of arithmetic, they are defined to be in the same sky if there exists a function f from N to N such that f(m)>n, and are in the same constellation if there are functions g and h such that g(m)=n and h(n)=m. The paper asks how many skies and constellations there are in M. The answers depend on properties, studied in a different context by Gustave Choquet, of the ultrafilter used to make M.